NASA Contractor Report 158947
ON SEVERAL ASPECTS AND APPLICATIONS OF THE MULTIGRID METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS
NATHAN DINAR
COLLEGE OF WILLIAM AND MARY WILLIAMSBURG VA 23185
NASA CONTRACT NAS1-14972-3 (NASA-CR-158947) ON SEVERAL ASPECTS AND APPLICATIONS OF THE MUTIGRID METHOD FOR
1118-32783
SOVTNG PARTIAL DIFFERENTIAL EQUATIONS (College of William and Mary) 58 p HC A04NF AO1 CSC
SEPTEMBER -1978 12A G364
--
Unclas 31570
IASA National Aeronautics and Space Administration
Langley Research Center Hampton Virginia 23665
httpsntrsnasagovsearchjspR=19780024840 2020-07-28T011819+0000Z
TABLE OF CONTENTS
Page
I Introduction 1
II Multigrid Methods for Systemsof Equations 4
21 Cauchy-Riemann Equations 522 Stokes Equations 923 Navier-Stokes 12
III Control and Prediction Techniquesin Multigrid Methods 16
IV Multigrid Methods for Time DependentParabolic Equations 21
V Improvements and Changes in ExistingMultigrid Algorithms 27
51 The Poisson Equation 2752 Distribution Relaxation 33
VI References 39
VII Appendix 40
ON SEVERAL ASPECTS AND APPLICATIONSOF THE MULTIGRID METHOD FOR SOLVING PARTIAL
DIFFERENTIAL EQUATIONS
By Nathan Dinar
College of William amp MaryWilliamsburg Virginia 23185
Research Sponsored by NASA Langley Research CenterContract NASl-14972-3
SUMMARY
Several aspects of multigrid methods are briefly described in this report The main subjects include the development ofvery efficient multigrid algorithms for systems of ellipticequations (Cauchy-Riemann Stokes Navier-Stokes) as well asthe development of control and predictiontools (based on localmode Fourier analysis) used to analyze check and improve thesealgorithms
Preliminary research on multigrid algorithms for timedependent parabolic equations is also described This reportdeals also with improvements in existing multigrid processesand algorithms for elliptic equations
Some partial and typical results are given More completeand detailed information will be presented in the authorsPhD Thesis to appear at the Weizmann Institute of ScienceRehovot Israel
I INTRODUCTION
This report deals with several aspects concerning multigrid
methods for fast solution of partial differential equations It
covers the research on this subject for the period August 1977 -
August 1978 when the author was spending his sabbatical at the
NASA Langley Research Center This research is part of a PhD
Research Assistant Department of Mathematics
2
Thesis to appear shortly at the Weizmann Institute of Science
Rehovot Israel including more detailed results and conclusions
The research on multigrid methods began in the early 1970s
by the initiative of Professor A Brandt He is today supervising
several projects on these subjects including the present research
The multigrid method can be applied to a wide range of
problems and therefore it interests many people It is known
to be one of the most powerful and advanced methods used today
The multigrid method uses the fact that the numerical
discrete equations we usually want to solve are not independent
They are derived from a continuous problem whose solution we
want to approximate In the process of the solution of the
discrete equations it is convenient to keep in mind the differshy
ential origin of the problem The use of discrete operators
on several levels of meshes interacting strongly in the process
of the solution allows us to solve the problems on the finest
grid very efficiently with a minimum number of arithmetical
operations (0(N)) where N is the number of equations in the
the finest grid
The multigrid method consists generally of several
processes performed in a given order and defining an itershy
ative cycle These processes include generally
Relaxation (Usually of Gauss-Seidel type) Used only to
smooth the errors ie to reduce these high
frequencies in the error that are not described
in coarser grids
3
Transfer of Residuals to a Coarse Grid This allows us to
define a problem on a coarse grid that is
similar to the original one The solution on
the coarse grid is used to improve the approxishy
mate solution given on the finer grid
Interpolation Used to define a new approximation on a
finer grid given an approximation solution on
a coarse grid
Moreover we can intermix the principles of the multishy
grid method with the grid adaptivity principles which mean
adaption of the order of discrete approximations and mesh
size using total smoothness of the solution This keeps
N as small as possible
These basic ideas and others as well as treatment of
theoretical and practical aspects have been extensively
covered in the papers of Brandt I2 3
This report deals with a-wide spectrum of problems
involved in the development of multigrid methods and multishy
grid software and the principal parts are the following
a) Development of multigrid methods and software
for elliptic systems of equations (Including
Cauchy-Riemann equation Stokes equation and
Navier-Stokes equation
b) Development of control and prediction tools for
various steps and processes involved in the
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
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6316 7316 8316
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3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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IC_ 6 l b1
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F-17 f711 E- 7 2F 7F-0ks
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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-shy I E On p6tlE~ont-shy
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183E-01 aonE-02
I 3
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59F+00
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-
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17J+00
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17CF-04
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a a
31 (1660 31 12 AhlF+011
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4 3
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5 a
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01
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37328
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shy I62EA I
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
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0F-a06 lQOt-051-0E -2E0
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
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3 3
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
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H iMlt+oo IhAE+on
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--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
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biiE~fl 177E4nP
0b77k-0l 28 E-0I
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A726i13 2t)E-03 htlt-03 QiAEtu
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shy
r 5
31APO 39070
IMSE+o l14I7p
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6 6
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1AqitIP03E+no
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21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
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423-E-02 -
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3 3
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
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9AhE01 031F02 3276fl
lqE-05- 1 06E
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-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
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PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
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1515
TABLE OF CONTENTS
Page
I Introduction 1
II Multigrid Methods for Systemsof Equations 4
21 Cauchy-Riemann Equations 522 Stokes Equations 923 Navier-Stokes 12
III Control and Prediction Techniquesin Multigrid Methods 16
IV Multigrid Methods for Time DependentParabolic Equations 21
V Improvements and Changes in ExistingMultigrid Algorithms 27
51 The Poisson Equation 2752 Distribution Relaxation 33
VI References 39
VII Appendix 40
ON SEVERAL ASPECTS AND APPLICATIONSOF THE MULTIGRID METHOD FOR SOLVING PARTIAL
DIFFERENTIAL EQUATIONS
By Nathan Dinar
College of William amp MaryWilliamsburg Virginia 23185
Research Sponsored by NASA Langley Research CenterContract NASl-14972-3
SUMMARY
Several aspects of multigrid methods are briefly described in this report The main subjects include the development ofvery efficient multigrid algorithms for systems of ellipticequations (Cauchy-Riemann Stokes Navier-Stokes) as well asthe development of control and predictiontools (based on localmode Fourier analysis) used to analyze check and improve thesealgorithms
Preliminary research on multigrid algorithms for timedependent parabolic equations is also described This reportdeals also with improvements in existing multigrid processesand algorithms for elliptic equations
Some partial and typical results are given More completeand detailed information will be presented in the authorsPhD Thesis to appear at the Weizmann Institute of ScienceRehovot Israel
I INTRODUCTION
This report deals with several aspects concerning multigrid
methods for fast solution of partial differential equations It
covers the research on this subject for the period August 1977 -
August 1978 when the author was spending his sabbatical at the
NASA Langley Research Center This research is part of a PhD
Research Assistant Department of Mathematics
2
Thesis to appear shortly at the Weizmann Institute of Science
Rehovot Israel including more detailed results and conclusions
The research on multigrid methods began in the early 1970s
by the initiative of Professor A Brandt He is today supervising
several projects on these subjects including the present research
The multigrid method can be applied to a wide range of
problems and therefore it interests many people It is known
to be one of the most powerful and advanced methods used today
The multigrid method uses the fact that the numerical
discrete equations we usually want to solve are not independent
They are derived from a continuous problem whose solution we
want to approximate In the process of the solution of the
discrete equations it is convenient to keep in mind the differshy
ential origin of the problem The use of discrete operators
on several levels of meshes interacting strongly in the process
of the solution allows us to solve the problems on the finest
grid very efficiently with a minimum number of arithmetical
operations (0(N)) where N is the number of equations in the
the finest grid
The multigrid method consists generally of several
processes performed in a given order and defining an itershy
ative cycle These processes include generally
Relaxation (Usually of Gauss-Seidel type) Used only to
smooth the errors ie to reduce these high
frequencies in the error that are not described
in coarser grids
3
Transfer of Residuals to a Coarse Grid This allows us to
define a problem on a coarse grid that is
similar to the original one The solution on
the coarse grid is used to improve the approxishy
mate solution given on the finer grid
Interpolation Used to define a new approximation on a
finer grid given an approximation solution on
a coarse grid
Moreover we can intermix the principles of the multishy
grid method with the grid adaptivity principles which mean
adaption of the order of discrete approximations and mesh
size using total smoothness of the solution This keeps
N as small as possible
These basic ideas and others as well as treatment of
theoretical and practical aspects have been extensively
covered in the papers of Brandt I2 3
This report deals with a-wide spectrum of problems
involved in the development of multigrid methods and multishy
grid software and the principal parts are the following
a) Development of multigrid methods and software
for elliptic systems of equations (Including
Cauchy-Riemann equation Stokes equation and
Navier-Stokes equation
b) Development of control and prediction tools for
various steps and processes involved in the
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCEFF 558 ACCEFF 557
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_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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1515
ON SEVERAL ASPECTS AND APPLICATIONSOF THE MULTIGRID METHOD FOR SOLVING PARTIAL
DIFFERENTIAL EQUATIONS
By Nathan Dinar
College of William amp MaryWilliamsburg Virginia 23185
Research Sponsored by NASA Langley Research CenterContract NASl-14972-3
SUMMARY
Several aspects of multigrid methods are briefly described in this report The main subjects include the development ofvery efficient multigrid algorithms for systems of ellipticequations (Cauchy-Riemann Stokes Navier-Stokes) as well asthe development of control and predictiontools (based on localmode Fourier analysis) used to analyze check and improve thesealgorithms
Preliminary research on multigrid algorithms for timedependent parabolic equations is also described This reportdeals also with improvements in existing multigrid processesand algorithms for elliptic equations
Some partial and typical results are given More completeand detailed information will be presented in the authorsPhD Thesis to appear at the Weizmann Institute of ScienceRehovot Israel
I INTRODUCTION
This report deals with several aspects concerning multigrid
methods for fast solution of partial differential equations It
covers the research on this subject for the period August 1977 -
August 1978 when the author was spending his sabbatical at the
NASA Langley Research Center This research is part of a PhD
Research Assistant Department of Mathematics
2
Thesis to appear shortly at the Weizmann Institute of Science
Rehovot Israel including more detailed results and conclusions
The research on multigrid methods began in the early 1970s
by the initiative of Professor A Brandt He is today supervising
several projects on these subjects including the present research
The multigrid method can be applied to a wide range of
problems and therefore it interests many people It is known
to be one of the most powerful and advanced methods used today
The multigrid method uses the fact that the numerical
discrete equations we usually want to solve are not independent
They are derived from a continuous problem whose solution we
want to approximate In the process of the solution of the
discrete equations it is convenient to keep in mind the differshy
ential origin of the problem The use of discrete operators
on several levels of meshes interacting strongly in the process
of the solution allows us to solve the problems on the finest
grid very efficiently with a minimum number of arithmetical
operations (0(N)) where N is the number of equations in the
the finest grid
The multigrid method consists generally of several
processes performed in a given order and defining an itershy
ative cycle These processes include generally
Relaxation (Usually of Gauss-Seidel type) Used only to
smooth the errors ie to reduce these high
frequencies in the error that are not described
in coarser grids
3
Transfer of Residuals to a Coarse Grid This allows us to
define a problem on a coarse grid that is
similar to the original one The solution on
the coarse grid is used to improve the approxishy
mate solution given on the finer grid
Interpolation Used to define a new approximation on a
finer grid given an approximation solution on
a coarse grid
Moreover we can intermix the principles of the multishy
grid method with the grid adaptivity principles which mean
adaption of the order of discrete approximations and mesh
size using total smoothness of the solution This keeps
N as small as possible
These basic ideas and others as well as treatment of
theoretical and practical aspects have been extensively
covered in the papers of Brandt I2 3
This report deals with a-wide spectrum of problems
involved in the development of multigrid methods and multishy
grid software and the principal parts are the following
a) Development of multigrid methods and software
for elliptic systems of equations (Including
Cauchy-Riemann equation Stokes equation and
Navier-Stokes equation
b) Development of control and prediction tools for
various steps and processes involved in the
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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IjIM111 VAU S- y +y(1y23) PC)
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l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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Al71E l P( __ AP -I tm I
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
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_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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9~ -it-0O5F rta
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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-- I7E+n3 - 00IF-I
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90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
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2 7 4 +0 I
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6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
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1515
2
Thesis to appear shortly at the Weizmann Institute of Science
Rehovot Israel including more detailed results and conclusions
The research on multigrid methods began in the early 1970s
by the initiative of Professor A Brandt He is today supervising
several projects on these subjects including the present research
The multigrid method can be applied to a wide range of
problems and therefore it interests many people It is known
to be one of the most powerful and advanced methods used today
The multigrid method uses the fact that the numerical
discrete equations we usually want to solve are not independent
They are derived from a continuous problem whose solution we
want to approximate In the process of the solution of the
discrete equations it is convenient to keep in mind the differshy
ential origin of the problem The use of discrete operators
on several levels of meshes interacting strongly in the process
of the solution allows us to solve the problems on the finest
grid very efficiently with a minimum number of arithmetical
operations (0(N)) where N is the number of equations in the
the finest grid
The multigrid method consists generally of several
processes performed in a given order and defining an itershy
ative cycle These processes include generally
Relaxation (Usually of Gauss-Seidel type) Used only to
smooth the errors ie to reduce these high
frequencies in the error that are not described
in coarser grids
3
Transfer of Residuals to a Coarse Grid This allows us to
define a problem on a coarse grid that is
similar to the original one The solution on
the coarse grid is used to improve the approxishy
mate solution given on the finer grid
Interpolation Used to define a new approximation on a
finer grid given an approximation solution on
a coarse grid
Moreover we can intermix the principles of the multishy
grid method with the grid adaptivity principles which mean
adaption of the order of discrete approximations and mesh
size using total smoothness of the solution This keeps
N as small as possible
These basic ideas and others as well as treatment of
theoretical and practical aspects have been extensively
covered in the papers of Brandt I2 3
This report deals with a-wide spectrum of problems
involved in the development of multigrid methods and multishy
grid software and the principal parts are the following
a) Development of multigrid methods and software
for elliptic systems of equations (Including
Cauchy-Riemann equation Stokes equation and
Navier-Stokes equation
b) Development of control and prediction tools for
various steps and processes involved in the
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
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OUTPUT No 3
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TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
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END CYCLE NO tshy
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-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
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2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
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OUTPUT NO 4IO LAO
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1515
3
Transfer of Residuals to a Coarse Grid This allows us to
define a problem on a coarse grid that is
similar to the original one The solution on
the coarse grid is used to improve the approxishy
mate solution given on the finer grid
Interpolation Used to define a new approximation on a
finer grid given an approximation solution on
a coarse grid
Moreover we can intermix the principles of the multishy
grid method with the grid adaptivity principles which mean
adaption of the order of discrete approximations and mesh
size using total smoothness of the solution This keeps
N as small as possible
These basic ideas and others as well as treatment of
theoretical and practical aspects have been extensively
covered in the papers of Brandt I2 3
This report deals with a-wide spectrum of problems
involved in the development of multigrid methods and multishy
grid software and the principal parts are the following
a) Development of multigrid methods and software
for elliptic systems of equations (Including
Cauchy-Riemann equation Stokes equation and
Navier-Stokes equation
b) Development of control and prediction tools for
various steps and processes involved in the
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
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4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
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3
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659L400
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31105
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17J+00
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13A E+Wl
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oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
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shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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02f-f5
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3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
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-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
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PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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----
---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
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I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
4
multigrid method (Based on local mode Fourier
analysis)
c) Development of multigrid methods for time
dependent parabolic equations (Heat equation
on rectangular domains)
d) Improvement and changes in existing multigrid
processes and algorithms (mainly on Poisson
equation) and treatment of new ideas in relaxation
methods
II MULTIGRID METHODS FOR SYSTEMS OF EQUATIONS
The basic ideas of the multigrid method are not
restricted of course to a unique equation and from a
theoretical point of view no special problem could be
expected in implementing multigrid ideas to a system of
equations However special and detailed algorithms for
this problem did not exist and it was extremely important
to get sharp and practical proofs of the efficiency of
multigrid methods for systems of equations One of the
questions we did not know the answer in advance was for
instance what is the appropriate method of relaxation for
a system
In order to answer this and many other important
questions we developed multigrid algorithms for three
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
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395E-02 775E-02 52TE-02 -
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1249E-03
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5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
--
-
3 __37_E 37E-2 3)6E-AA--------IO7 E - - 3 _ 845-_ 7U E-I)1t_ 1- 9E-0Q 7 I3 F-)7
____ ____
3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
70IE+ol PNOE-o
7AI -OU
3
2
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303ia
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6137E-llO
IP~E-nlshy
thE+1)O
659L400
qalE+O01
SakE-00
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3
3 3shy
31105
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17J+00
jF4(MO SAIE-nI
13A E+Wl
ahF41 shy- I(S -ol+0
oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
- - PA7E-O 3 7E-0
pqPE-0)5 63PE- 3QOE--
02f-f5
-shy
- --- ---
3 3
37114 7U3)
3ql -n5Eshy$nFtp IIA-tl
- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
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----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
5
typical problems starting with a simple one and each
subsequent problem involving more and new complications
relative to the former Moreover since it is usually
wise and necessary in this kind of research to isolate
different questions which may arise we developed the
algorithm for simple geometrical domains (rectangles)
The past experience showed that more complicated geometries
did not affect the efficiency of multigrid methods (see
for instance Shiftan4) For complicated geometries there
are of course more programing problems
21 Cauchy-Riemann Equations
The equations are
Sx-Vy fl (xy) (i)
Ux+Vy = f3(xY) (2)
for 05xsl Osyll
And the boundary conditions
U(Oy) = U0 (y) V(xO) = V0 (x) (3)
U(ly) = Ul(y) V(xl) = Vl(x)
Figure 1
y V (x)
U (y) iU(y)
v0 )x)
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
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4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
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6316 7316 8316
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
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2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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37361 3717 373Q5 37398 -
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
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3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
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U0hi-Ip 21qEnp
622Ff-)
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- 236E-02 I41(T-A6723b-n6
------shy
----
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41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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-- - - - -
---
-- --- ---
--- -
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----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
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S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
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-
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
6
The data must fulfill an additional condition due to
the continuity equation (2) in order to assure-a solution
to the system This condition is
0V fIvj- 0 )dx f3 y)dxdy(I c 0
The discretization of the problem was accomplished
on a staggered grid This method has several advantages
in problems of this type For instance it allows us to
define easily second order conservative finite differences
The grid is described in figure 2a
Figure 2a
VVkV
U U2 f3 J3 U
U U V U
V V V
The finite differences approximations for equation-()
are defined at grid intersections (X) for instance
71 P jJ3t 1-(v2 V1 J
And the finite differences for equation (2) are defined in
the middle of the cells () ie
1u 3 -U 2 +V3 vk = f3 (6)
As in the continuous case the data must fill an additional
discrete condition equivalent to (4) in the finest grid ie
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
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3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
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1249E-03
T170E-03 130E-03
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13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
- q7 oR-o p lE-p 1W 1 - --- shy-hq2E-n5 JlE-IoS
9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
__~ ~A P3 30) iflOE-rll I nIIE-All - AEf5ARE41S -- FN CY NJ) shy ~~ ~ I ___--- I)5IpE-nfj I 01-tS|ltF114 IIhF-tIII
____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
_ I -bS4 -
2ho17 933
s~~lq l|dF-iiS u1E53
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PPF-1 I- r-n5
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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3)-I7E-05 211 E-rIS_ -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
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IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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00
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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183E-01 aonE-02
I 3
3nw 3n677
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I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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3lhF01
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31105
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17J+00
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17CF-04
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a a
31 (1660 31 12 AhlF+011
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4 3
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01
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37328
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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2B31)E-IIOEkfo S7E-(
367F-(2
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
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biiE~fl 177E4nP
0b77k-0l 28 E-0I
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31APO 39070
IMSE+o l14I7p
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- 1IAQE-03 SE-03
6 6
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1AqitIP03E+no
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21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
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sp5E-oll 7QE-02
262P-ll 1
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1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
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423-E-02 -
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
I I -01 AAOEnh 12116-Os
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3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
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----
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41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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e wEND CYCLE NO
-
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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p-Io0 L
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I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
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------- -~~- -0- - -- -5- - N
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CYCLE NO 1i
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- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
7
(Ulj1 -U 0 j)+ L (Vii-Voi) = J f3ij (4a)
Several relaxation schemes appropriate to this system
were considered and all these schemes have a good smoothing
factor around 7=5 The most convenient scheme was chosen
because of its simplicity and its small number of arithmetical
operations This method belongs to a new approach of relaxshy
ations developed by Professor A Brandt and is called
Distributed Relaxations
This approach is characterized by the fact that when
passing through a point (or cell) in the grid for which
the difference equations are defined we change the value
of more than one unknown in order to make zero residual
on this point
In the Cauchy-Riemann equations separate relaxations
are performed to each one of the equations according to
the schemes discribed in figures 3 and 4
Efficiency of relaxation in a multigrid process is measuredby the smoothing factor which is defined as the asymptoticratio between the fast components of the errors after a relaxshyation sweep and the same errors before the relaxation A more
exact definition can be found in Brandt 2 More considerationson this subject can be found in Chapter 3 of this report
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
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3-764E-03 107E-02
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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128E-03 E108E03135E-03
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
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1249E-03
T170E-03 130E-03
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
- q7 oR-o p lE-p 1W 1 - --- shy-hq2E-n5 JlE-IoS
9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
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Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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59F+00
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I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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17CF-04
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31 (1660 31 12 AhlF+011
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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3 a3 at)1I 13tl
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-
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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I-COZZZ 0000
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~~~ -
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V0
---- -C -0 c0
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------- -~~- -0- - -- -5- - N
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CYCLE NO 1i
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- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
8
Figure 3 Relaxation of Ux-V = f1
u2+
C
Figure 4 Relaxation of Ux+Vy
2
4f3 U-1i-+
The values U+j0 and V+ C in figures 3 and 4 represents the
new values after the sweep of this point 0 is always chosen
to make the residual zero at the point The relaxation is a
Gauss-Seidel type relaxation and it passes through all the points
(or cells) of the grid in a usual natural order The important
property of this method relies on the fact that when relaxing
each of the equations the residual of the other equation
remains unchanged
Mode Fourier Analysis shows that the smoothing factor for
this method is i =5 the same as in Poisson equation which
is of course equivalent to the Cauchy-Riemann system The
multigrid algorithm shows convergence factor 7 56 which is about the same value as in the Poisson case The
number of operations in the relaxation of the Cauchy-Riemann
See Chapter 3 in this report
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
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-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
- q7 oR-o p lE-p 1W 1 - --- shy-hq2E-n5 JlE-IoS
9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
__~ ~A P3 30) iflOE-rll I nIIE-All - AEf5ARE41S -- FN CY NJ) shy ~~ ~ I ___--- I)5IpE-nfj I 01-tS|ltF114 IIhF-tIII
____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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IC_ 6 l b1
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F-17 f711 E- 7 2F 7F-0ks
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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00
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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59F+00
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I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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31 (1660 31 12 AhlF+011
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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3 a3 at)1I 13tl
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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~~~ -
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CYCLE NO 1i
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1515
9
system is of course higher than in Poisson equations (about
two times more) On the other hand we can say that the
information we get from the solution of the Cauchy-Riemann
system (functions U and V) is also two times the information
we get from the solution of the Poisson equation
The main reason for our treatment of Cauchy-Riemann
system was for learning purposes It is probably the simplest
elliptic system possible and it allows us to treat complishy
cated problems more easily However this system is intershy
esting in itself as we can see in the works of Lomax5 and
Ghil6 These papers describe fast algorithms for the
solution of the Cauchy-Riemann system We think that the
algorithm just described in this report is preferable
because it does not depend in principle on the simplicity
of the domain and because it is at least as fast as Chil
algorithm and even faster asymptotically (for finer and
finer discretizations) The implementation of the algorithms
for nonlinear equation is also much easier in the multigrid
3method
An example of a computer output is given in the Appendix
Output No 1
22 Stokes Equations
The Stokes equation are given by
AU-px = f (7)
t1 V-py = f2 (8)
U +V = f3 (9)
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
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3-764E-03 107E-02
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8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
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1249E-03
T170E-03 130E-03
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13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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E-14 l3-d i - IM3A II - 19 fqJ-Cu6 - lbI-
1 IJ bbl E-U t U E -I -
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy
-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
-06 shy
S- F- 5 1E-I5 - RQ1U-s 4AI
- q7 oR-o p lE-p 1W 1 - --- shy-hq2E-n5 JlE-IoS
9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
__~ ~A P3 30) iflOE-rll I nIIE-All - AEf5ARE41S -- FN CY NJ) shy ~~ ~ I ___--- I)5IpE-nfj I 01-tS|ltF114 IIhF-tIII
____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
_ I -bS4 -
2ho17 933
s~~lq l|dF-iiS u1E53
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PPF-1 I- r-n5
h 87Fshy 3 t9R _P 7rE- I)h -uaIP- Io- --2F--hb F - - - - - - - - - --- 2 __ PbR I 4E-117 bt3E-A 7 3QJP -t07 1 I 0F-06h
_3 I
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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3)-I7E-05 211 E-rIS_ -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
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15deg90 2 1
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IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
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183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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3lhF01
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659L400
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3 3shy
31105
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17J+00
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13A E+Wl
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1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
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7021 -01 SM -flil
5 a
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01
2 2
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37328
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shy I62EA I
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
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0F-a06 lQOt-051-0E -2E0
-
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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2B31)E-IIOEkfo S7E-(
367F-(2
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3 3
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
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H iMlt+oo IhAE+on
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--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
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biiE~fl 177E4nP
0b77k-0l 28 E-0I
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A726i13 2t)E-03 htlt-03 QiAEtu
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shy
r 5
31APO 39070
IMSE+o l14I7p
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- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
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21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
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423-E-02 -
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3 3
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
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9AhE01 031F02 3276fl
lqE-05- 1 06E
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
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PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
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----
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41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
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e wEND CYCLE NO
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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p-Io0 L
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I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
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------- -~~- -0- - -- -5- - N
lz 11 1
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0 b
1
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CYCLE NO 1i
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- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
10
The unknowns are U V and P U and V are given on
boundaries Like in Cauchy-Riemann systems The data must
satisfy the condition (4) The discretization is also in the
same staggered grids already described P and f3 are defined
in the centers of the cells f1 and f2 are defined in the
same points as U and V respectively (see figure 2b)
Figure 2b
V-V -v v
U P U P U P U PII
U P U P U P U P
I -V-u- I --V- 0 -- -V I- VVI_
The difference equations for equation (7) are given by
AucxtxlampiP (x+~y)-P(x- S Y)] f(xy) (10)
for (xy) where U is defined A is the usual five
points discrete Laplace operator Li -
Similarly for equation (8)
I CxY+ )-P(xy- ff2 (xy) (1i)(xxyY)- )
for (xy) where V is defined
For the continuity equation (9) the difference
equations were already defined by (6)
Equations (10) and (11) hold for interior points Near
the boundary the difference equations are different but they
are simple and will not be described here
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
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6316 7316 8316
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3-764E-03 107E-02
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8977 _8 PO
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1983-_ 11984 -shy - - - --
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5 940E-04 249E004 9364E-04 34816
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4 5 6 6
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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00
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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183E-01 aonE-02
I 3
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59F+00
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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a a
31 (1660 31 12 AhlF+011
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4 3
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37328
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shy I62EA I
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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367F-(2
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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1AqitIP03E+no
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21gt~E+02 237E+MO -
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h 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
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-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
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U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
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41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
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50 1 PS3E -( I t
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2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
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Cr wt
-- - - - - -
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
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S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
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S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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-
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
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- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
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-- - - - -- - - - Z az tI-Z3zz
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----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
11
The relaxations are based on the same principles already
described The relaxation of equations (10) and (11) the
Momentum equations are performed separately by the usual
Gauss-Seidel method corresponding to the Poisson operator
The relaxation scheme for the continuity equation (6) is
described in figure 4
Figure 4II
ampis chosen so that equation (6) is fully satisfied on the
current cell and the changes of the values of the P are
chosen in a way that preserves the residuals of equations
(10) and (11) when relaxing equation (6)
The theoretical smoothing factor in the relaxation
method just described can be shown to be I =5 like in
Poisson and Cauchy-Riemann equations
In the numerical experiment we get a multigrid convershy
gence factor - 65 This is a very good value and assures
This is exact only on unbounded domains On bounded domainsthis property cannot be strictly maintained near the boundary
This is the reduction factor of the residuals per unit workwhich is equivalent to one relaxation sweep on the finest grid
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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6316 7316 8316
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
___
___ _ 6 Pi PAORJ
3)-I7E-05 211 E-rIS_ -
-1j-1307V-ol
Ok~tF-I qshyE-hIIF-tI4
~ ~ -
-
E -
___ S - 7a - __ r -- shy---------shy - -shy shy
-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
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3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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3lhF01
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3
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659L400
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3 3shy
31105
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17J+00
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oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
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762E+01
95UE+00 37nE-0I 4O O
4 3
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pj]lon161)F4041
7021 -01 SM -flil
5 a
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01
2 2
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37328
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shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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02f-f5
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- --- ---
3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
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H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
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Sh =3 xEn
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
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- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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CYCLE NO 1i
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- CYCLE N0-O-CYCLE EFF i CCFF 66A
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1515
12
a very fast method for solving the Stokes equations However
it was a little worse than our expectations based on the
previous experience of Poisson and Cauchy-Riemann equations
In order to check this point and to reduce the possishy
bility of errors in the computer program we performed a
complete multigrid Mode Fourier Analysis (with the aid of the
techniques to be described later) The results of this analysis
showed complete agreement between numerical and theoretical
results leading to the conclusion that the value of 65
is intrinsic to this system which is characterized by strong
interaction between the equations The Mode Fourier Analysis
helped to rule out possible programing errors or bad influences
of the boundary on interior regions
An example of a computer output is given in the Appendix
Output No 2
23 Navier-Stokes Equations
Two-dimensional Navier-Stokes equations are given by
[3x(12)
LxR~~x1 1 Vtj~- 21)(13)
)X C (14)
and the same boundary condition as in Stokes equation The
parameter R is called the Reynolds number (R=O corresponds
to the Stokes equation)
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
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6316 7316 8316
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3-764E-03 107E-02
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1249E-03
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
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3
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303ia
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659L400
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3 3shy
31105
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17J+00
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13A E+Wl
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oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
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shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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02f-f5
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3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
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-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
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-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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---
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----
---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
13
These equations are more interesting from the practical
and physical point of view than the former equations The
numerical treatment is more difficult and complicated due
to several reasons like the nonlinearity and the fact that
boundary-layers may appear for large Reynolds numbers
These facts demand a careful choice of the difference
equations in order to keep the ellipticity of the difference
operator This can be accomplished in several ways based
usually on one-sided first differences instead of central
first differences that may cause instability unless we make
a very drastic and unpractical restriction on the mesh size h
1(h must be of the order h=O(K)) In this work we define
the following difference approximations to the first derivatives
of U and V for example for we define
x- 0-a0 definULe (15)
for 0ltaxpl (ax = 5 corresponds to central differences)
a is defined byx
_i )(16)
I-- 1
21 = LRU(xy)and
U gtVan - aedfndi The approximation for and - are defined in
a similar way It can be shown that with this approximations
we get finite differences with the desired properties The
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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15_ 1 IVnE4+110 I EIf I -n I E4 qli5 3 IE4 l IIEI R_E~f l FSi Ell 110A ktE4Ii
l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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Al71E l P( __ AP -I tm I
P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3
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1I 11~75 431
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9hlE-13 AhfF-v3 1AfE-n3
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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9~ -it-0O5F rta
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
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2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
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2 7 4 +0 I
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
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3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
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4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
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5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
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5 b
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5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
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50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
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-
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- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
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0 b
1
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
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II 5E-fl2 - oqiAE-OM 279E05
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
14
equations defined on the same staggered grid already
described are given by
MK~ue1) - P F(tI (8
(19)
(20)
(21
-k~U lRk (22)
4 4shyc-+(23)
(These are the equations for interior points The operator
is a little different near boundaries)
The relaxation method for (18) and (19) are done as in the
Stokes equation by freezing U and V in the calculation of
(9(1) and N)
The relaxation scheme for equation (20) is described in
figure 5
ORM1INAL PAGE ISQ ooR QUALM
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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-- I7E+n3 - 00IF-I
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6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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1515
ORIGINAL PAGE IS OF POOR QUALITY
115
Figure 5
Vt C
is chosen so that the residual of the continuity equation
in the current cell is zero The changes in P wouid keep
the residuals on (18) and (19) unchanged if we froze the
coefficients
Mode Fourier Analysis shows that the smoothing
factor (calculated for equations with frozen coefficients)
will generally depend on the direction of the relaxation of the
Momentum equations For instance if the solutions U and V
satisfy U)O VG and the direction of the relaxation is as
usual (say by columns and increasing y and increasing x)
then 5 If U or V are negative in some part of the
domain and R is big enough the smoothing factor will be
higher and therefore the multigrid method less efficient
These difficulties may be easily overcome by making relaxations
sweeps in different directions improving in this way the
smoothing factor or better by the technique of distributed
relaxations already mentioned
The first preliminary version of the algorithm does not
include the features just pointed out and therefore it works
16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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17CF-04
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
5 b
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sp5E-oll 7QE-02
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16
with the full multigrid efficiency only for cases where U and
V do not change sign in the domain or for general cases with
R not too big (up to R=100)
These first experiments are very important in showing
how the algorithms work in principle independent of secondary
problems like the direction of the relaxations mainly
technical
The numerical results shows a good multigrid convergence
factor bounded for all R by 7-75 This provides
a very fast algorithm for solving Navier-Stokes and
almost the same efficiency as in the linear Stokes case
The full multigrid cycle (explained later) was also
implemented Preliminary experiments shows that only
8-9 works units are needed to solve the equations
to the level of truncations errors (Execution time
on CDC 6400 for a grid of 64x64 is about 15 seconds
and the program is not optimized and includes all kind
of calculations for debugging purposes) An example
of a computer output is given in the Appendix Output
No 3
III CONTROL AND PREDICTION TECHNIQUESIN MULTIGRID METHODS
One of the remarkable advantages of the multigrid
method is the possibility of predicting in advance its
efficiency given the relaxation method interpolation
Equivalent to relaxations sweeps on the finest grid
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
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6316 7316 8316
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3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1249E-03
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5 940E-04 249E004 9364E-04 34816
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4 5 6 6
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
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15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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-shy I E On p6tlE~ont-shy
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183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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3lhF01
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31105
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17J+00
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17CF-04
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a a
31 (1660 31 12 AhlF+011
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4 3
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01
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37328
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shy I62EA I
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
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0F-a06 lQOt-051-0E -2E0
-
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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2B31)E-IIOEkfo S7E-(
367F-(2
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3 3
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
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H iMlt+oo IhAE+on
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- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
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biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
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- 1IAQE-03 SE-03
6 6
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1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
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E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
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-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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---
-- --- ---
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----
---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
17
etc This can be done by means of local Fourier Mode
Analysis in an infinite space ie neglecting boundaries
influence This can be justified because of the fact that
Fourier Analysis is very accurate in descibing the fast
components behavior while is less accurate for slower
components However most of the calculating work in the
multigrid processes is invested in the reduction of the
fast components of the errors in the finest grid and this
is achieved by means of relaxation More detailed justishy
fications can be found in Brandt 3 who shows a very good
agreement between theoretical results based on Fourier Mode
Analysis and between numerical data obtained by multigrid
algorithms
We can distinguish two levels of analysis that are
appropriate for multigrid methods In the simplest one
we calculate the smoothing factor 4 which depends only
on the relaxation process The more complete and complicated
analysis takes into account all the multigrid processes and
estimates theoretically the value of t (definitions of 7
and 4 are given in the footnotes in Chapter 2) In many
cases it is enough to perform the simpler analysis In - i i0
these cases the formula I = I wheref is the
ratio of the number of points between coarse approximate
and fine grid (usually S-) and d is the dimension of
the problem provides a fair approximation to
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
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128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
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1249E-03
T170E-03 130E-03
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
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o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
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131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
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3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
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4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
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3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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e wEND CYCLE NO
-
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
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1
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
18
The calculatibn of the smoothing rate involves the
search for extremum of functions in bounded domains and the
ca-cu-lation of etgenvalues (defined -shysrally by a gheralshy
ized eigenvalue problem Ax=ABx) of complex matrices of
size q9 where 4 is the number of equations in the
system The estimation of the convergence factor
involves the same kinds of calculations but the size
of the matrices are much bigger ( 2 x 2 where d
is the dimension of the problem)
It is clear that the kind of calculations just
described cannot be performed in a closed form
with the exception of extremely simple cases like the
smoothing rate for Poisson equations and one dimension
simple problems
Because of this and other important reasons an
algorithm that performs these calculations was constructed
A general FORTRAN subroutine was written for this purpose
and the main inputs are the following
d - The dimension of the problem
q - The number of differential equations = the
number of unknowns
r - The number of relaxations in one multigrid
iterative cycle
In the relaxation process the Fourier components areindependent But when we take into account the transferfrom grid h to grid 2h we get interdependence between
groups of 2d components
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
- -shy
- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
__
OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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_ S - no - IIFI - l7Ftn I3amphl LI CYCLE l
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- -E -I A 1 F - - A I f 1 - - - - - -n - shy
- to I Ef hQ Q $OE (I -- - shy
--shy t( Pl3( FQ FIII 110 02F
15_ 1 IVnE4+110 I EIf I -n I E4 qli5 3 IE4 l IIEI R_E~f l FSi Ell 110A ktE4Ii
l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
U-
-~ ~~ ~A~7 F +Icw7c A - 11n-F+ PiI)hf~ Izzz-----------nshy -shy - -shy -shy -
a 5 9tr 9
- ___
A0 Eshy 1o) 7E-nl 4J IE O1
5F-1 t 7 E-)IAIFP- Il
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S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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1515
19
AB - Complex matrices that are derived from the
particular difference equations and methods
of relaxations The size of these matrices
is A and theydepend on the Fourier
component defined by e=(SJampL )6A)
A A function of the Fourier component 0
describing the initial distribution of
amplitudes in the errors (For a random
initial distribution of errors A0 =i
for all amp)
L - Number of points defining a mesh in the ampdomain
S- The ratio between the number of points on grid
and the number of points on grid
The output contains mainly
a) The smoothing factor 4 defined by
4ncz-~(25)
where
d eE CA V- e0(26)
and JAIX1 b) The weighted smoothing factor j given
the number of relaxations r and the weight
function A amp
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1249E-03
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5 940E-04 249E004 9364E-04 34816
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4 5 6 6
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
--
-
3 __37_E 37E-2 3)6E-AA--------IO7 E - - 3 _ 845-_ 7U E-I)1t_ 1- 9E-0Q 7 I3 F-)7
____ ____
3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
70IE+ol PNOE-o
7AI -OU
3
2
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303ia
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6137E-llO
IP~E-nlshy
thE+1)O
659L400
qalE+O01
SakE-00
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3
3 3shy
31105
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17J+00
jF4(MO SAIE-nI
13A E+Wl
ahF41 shy- I(S -ol+0
oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
- - PA7E-O 3 7E-0
pqPE-0)5 63PE- 3QOE--
02f-f5
-shy
- --- ---
3 3
37114 7U3)
3ql -n5Eshy$nFtp IIA-tl
- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
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----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
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I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
20
c) The location (mode) where the smoothing
factor 7 is found
d) A distribution map of 10) (only for the
two-dimensional cases d=2)
The same subroutine with some changes can be used
for the complete Fourier Multigrid Analysis In this case
the user has considerably more work in defining the complex
matrices A and B that include in this case the transfer
of residuals from grid h to grid 2h and the interpolation
from grid 2h to grid h
These programs are very important in the research of
multigrid methods it can be used for several purposes
for example
1) Checking new relaxation methods or new multigrid
processes before the algorithm is translated into
the computer
2) Comparison of several algorithms for the purpose
of optimization ie looking for the faster and
stable ones
3) Debugging of multigrid computer programs
We used the subroutine in several cases including
Cauchy-Riemann and Stokes equation We also used it for the
Poisson equation In all these cases we did the complete
Fourier multigrid analysis checking and comparing various
methods of relaxation residual weighting and interpolation
We got full agreement between theoretical and numerical
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
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- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
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377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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OUTPUT NO 2 Yd
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
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END CYCLE NO tshy
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
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-
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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~~~ -
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CYCLE NO 1i
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--0
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1515
ORIGINAL PAGE IS OF POOR QUALITY
21
results (in all cases considered) and this fact increases
of course the reliability and prediction power of this
theory
An example of these facts are summarized on table 1 which
include theoretical and numerical results for Stokes equation
Table 1
Comparison Between Theoretical and Numerical Resultsfor Stokes Equation
(The number of Parameter related ultigrid Convergence Factor relaxations on to the degree of the finest grid accuracy in solving Theoretical multi- Numerical in one iterative coarse grid correctshy grid complete results multigrid cycle) ion equations Fourier analysis
1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722
The subroutine was also applied for checking a special approach
to distributed relaxations that may perhaps be applied to
Navier-Stokes equations This will be shortly described in
Chapter 5 An example of a computer output is given in the
Appendix Output No 4
IV MULTIGRID METHODS FOR TIME DEPENDENT
PARABOLIC EQUATIONS
A possible quite obvious way to use multigrid procedures
for initial value problems is considered in the work of
22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
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3-764E-03 107E-02
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8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
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T170E-03 130E-03
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13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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9~ -it-0O5F rta
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 3)907 I ft)I
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
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433 O5 357n I3n33
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sp5E-oll 7QE-02
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3 a3 at)1I 13tl
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I 3
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22
Brandt3 By this procedure we use a multigrid algorithm
for solving the implicit equations usually defined at each
time step If for instance we want to solve the Heat
equation in two space dimensions then for each time step
an elliptic problem similar to the discrete Poisson equation
is defined (and in this case the Gauss-Seidel relaxation
has even better smoothing properties than in the Poisson
case) The typical amount of work needed in advancing each
time step by multigrid procedures will be accordingly
equivalent to 5-6 relaxations (see 3) if a solution for
M time steps is required then the total amount of work
will be about 6M relaxations
The question which naturally arise is whether we can
use multigrid principles anI ideas to get more efficient
methods for these Problems
The answer appears to be positive and Professor A
Brandt pointed on some apnroaches which eventually can develop
in efficient multigrid algorithms for these problems
A basic idea is that marching in time can be done for
most of the time steps on coarser grids
The appropriate equations on a coarse grid are carefully
corrected in a way that assures the correct representation
of the information from the fine grid so that even when
marching on coarser grid the accuracy of the finer grid is
kept To update the information from the finer grid we need
some sporadic and infrequent time steps on the finest grid
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
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OUTPUT No 3
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TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
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END CYCLE NO tshy
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-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
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2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
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2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
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3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
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3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
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6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
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50 1 PS3E -( I t
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2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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- -
e wEND CYCLE NO
-
-
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----- -- - ------shy
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-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
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------- -~~- -0- - -- -5- - N
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1
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----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
ORIGINAL PAGE IS OF POOR QUALITY
23
that constitute of course most of the computational work
The base of this approach relies on the fact that fast
Fourier components of the solution (represented on finer
grids) converge to steady-state after a very short time
Changes in the solution after this are due mainly to slower
components that also change slower in time This allows
us to march on coarser grids with large time steps after
the influence of fast components have disappeared
In order to check these and other ideas we developed
two different algorithms for the Heat equation in a
rectangular domain with Dirichlet Boundary Conditions The
first algorithm uses the Crank-Nicholson implicit scheme
and the second one uses the simplest explicit scheme It
must be pointed out that the stability restriction in the
explicit scheme that makes this method very unpractical
does not appear in this case because most of the time we
march on very coarse grids where large time steps are allowed
a The Implicit Scheme
The equations to be solved on the finest grid
(with mesh size h0 and time step k0 ) are
J tL+Pe X (27)
and the unknowns are u0(xyt+k0) for each (xy) definedId
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
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IC_ 6 l b1
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F-17 f711 E- 7 2F 7F-0ks
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
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3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
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3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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183E-01 aonE-02
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I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
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I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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31 (1660 31 12 AhlF+011
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
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1515
24
the finest grid Similarly umhmkm will represent marching
on a grid which is m levels coarser The equations on coarser
grids are given by
Ut ~ ~ A ~ ~ shy S1L~~r
L(XA 7) (23)
-where usually and T represents the appropriate
correction -C has a very important significance and it
represents the spacial truncation error of grid m relative
to grid 0 TU is given by
A s (29)
where n s) satisfying S C
The transfer from a given coarse grid m to a finer grid
m-i is done by rn-i rn-i in-m rn-i
uULast + m u -u Last) (29a)
where lmm I means interpolation (cubic) from coarse m
to fine grid and Last renresent the last marching in
time on the m-i grid
b The Explicit Scheme
The equations on the finer grid are
jO Lt t )-t1l t - ixc y )= f~ xy) (30)
(
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
- -shy
- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
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--shy t( Pl3( FQ FIII 110 02F
15_ 1 IVnE4+110 I EIf I -n I E4 qli5 3 IE4 l IIEI R_E~f l FSi Ell 110A ktE4Ii
l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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_ 5 R _ __--shy SnwFI __ 4shy A 1n11 PIE-1 t -
Al71E l P( __ AP -I tm I
P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3
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1I 11~75 431
3 uS- 14
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9hlE-13 AhfF-v3 1AfE-n3
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P1-PE-flIS 1 71I1E-05 01 6EO04
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1114E-03 At~IIE-fI4
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
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26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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2ho17 933
s~~lq l|dF-iiS u1E53
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PPF-1 I- r-n5
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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3)-I7E-05 211 E-rIS_ -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
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15deg90 2 1
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32E-07 1 t -7-shy306E-OA
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
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--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
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2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
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--0
tWOn
1515
ORIGINAL PAGE IS 25OF POOR QUALITY
On coarser grids the following equations are defined
t~ (3 1)
In this algorithm we keep a constant ratio A over
all grids given by A 1 The value of A 2
was chosen as very appropriate from the point of
view of the fast Fourier components
represents
here the complete relative truncation error (in
space and time) ishyis defined as follows
For m=l
T S) L4txL sxt xy sV fx ) (32)
and (t4 Then for general m
j 1 k- (32a)
and s is defined as in (29) The transfer from coarse
to finer grids is performed like in (29a)
It must be pointed out that this research which deals
with new approaches in implementing multigrid ideas has
a very preliminary character and the principal aim was
to check potentially the various promising possibilities
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
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6316 7316 8316
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
__~ ~A P3 30) iflOE-rll I nIIE-All - AEf5ARE41S -- FN CY NJ) shy ~~ ~ I ___--- I)5IpE-nfj I 01-tS|ltF114 IIhF-tIII
____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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183E-01 aonE-02
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59F+00
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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a a
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01
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37328
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shy I62EA I
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777
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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367F-(2
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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21gt~E+02 237E+MO -
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h 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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IflQEtll
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
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-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
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622Ff-)
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- 236E-02 I41(T-A6723b-n6
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----
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41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
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50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
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-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
26
In this context some numerical experiments were performed
in which we tried some criteria in order to define the
times where we switch from coarse to finer grids and viceshy
versa In addition we compared the accuracy we got in these
algorithms with the same schemes when we marched in time on
the finest grid only The numerical results we got from
these experiments in both algorithms are good and encouraging
However the explicit scheme appears to be preferable in some
aspects The numerical results in these experiments and some
theoretical considerations arising from the interpretation
of these results lead to the conclusion that applying these
multigrid ideas with full efficiency (ie solving the problem
in an amount of work comparable with the work invested in
solving elliptic problems) means using adaptive techniques
where we can change and adapt the order of the discrete
approximation
But even without adaptive techniques we can use the
present algorithms to solve the Heat equation very efficiently
The amount of work needed will depend on the smoothness of the
solution For instance for a problem with smooth initial
conditions we can save 98 of the work in relation to the
same numerical scheme defined on the finest grid only (of
size 64x64)
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCEFF 558 ACCEFF 557
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OUTPUT NO 2 Yd
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
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OUTPUT NO 4IO LAO
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OPEATO 119ooooXAr
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1515
27
V IMPROVEMENTS AND CHANGES IN EXISTINGMULTIGRID ALGORITHMS
In this chapter we deal with existing multigrid proceshy
dures and we check various approaches in order to improve
their efficiency The first part deals with the practical
solution of Poissons equation in a small number of numershy
ical operations Although in this part we concentrate on
Poissons equation for reason of convenience when checking
and comparing numerical results it is quite clear that the
techniques developed here can be applied to other elliptic
problems as well
The second part of this chapter deals with a special
approach to distributed relaxations for general five points
difference operators of the form la -s c s = a+b+c+db
51 The Poisson Equation
The first numerical multigird algorithm and program
was written for the Poisson equation For this equation
there exists today more information related to multigrid
methods than for any other problem
In Brandt I the algorithm called there Cycle C
for solving the Poisson equation is described with detail
In Cycle C the multigrid iterative cycle starts at the
finest grid where an initial approximation to the solution
is defined This approximation is then improved by solving
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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6316 7316 8316
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3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
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3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1249E-03
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5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
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3
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659L400
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31105
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17J+00
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13A E+Wl
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oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
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762E+01
95UE+00 37nE-0I 4O O
4 3
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
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shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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02f-f5
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- --- ---
3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
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H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
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----
---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
--
28
correction equations on coarser grids The switching from
fine to coarse grid and vice versa is controlled by some
internal criterplusmna
Cycle C is very useful in learning and understanding
multigrid performance and several theoretical aspects
like the asymptotic convergence factor etc It is
not generally the most efficient algorithm for solving
real problems for which we do not know in advance a
good approximation on the finest grid In addition it
is usually difficult to know in advance how many iterative
cycles are needed to get the desired accuracy One can
perform several iterative cycles and then get very close
to the solution of the difference equations but this will
generally be wasteful because in a real problem we do
not need more accuracy than the accuracy defined by the
differential truncation error
Because of these and other considerations a fixed
algorithm for Poisson equation is described in Brandt3
In this algorithm we start with an approximation on the
coarsest grid and after we perform one iterative cycle
on each level the solution on each grid is used as a first
approximation on the next finer grid by means of a cubic
interpolation This algorithm is based on the knowledge
and past experience in solving Poisson equation by multishy
grid techniques and provides a solution to the problem
up to an accuracy comparable with the truncation errors
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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IjIM111 VAU S- y +y(1y23) PC)
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LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
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U-
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_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
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PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
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2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
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6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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e wEND CYCLE NO
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
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~~~ -
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---- -C -0 c0
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1
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CYCLE NO 1i
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--0
tWOn
1515
ORIGINAL PAGE IS OF POOR QUALITY 29
accuracy and this solution is found in a minimal number
of arithmetic operations (we do not even need to calculate
the residual norms used generally-forinternal criteria)
In the present work a similar but more general algorithm
is constructed This algorithm can serve for learning
purposes as well as for solving practical problems It can
also be easily applied to other elliptic operators The
algorithm was implemented in the FAS (Full Approximation
Storage) mode (For a detailed description of FAS see
Brandt2) In this mode we can look at the coarse grid
correction equations in a somehow differentpbut equivalent
way as follows2k z U
Tj~(~ Ljuk(33)
U2h is a new and better approximation than Uh to theI
exact solution Mh of the difference equation
L (34)
Zk T given by (33) describes the truncation error of the
grid 2h relative to grid h It is well known that the exact
solution u of the differential equation Lu-f satisfy the
following
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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2ho17 933
s~~lq l|dF-iiS u1E53
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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9~ -it-0O5F rta
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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-- I7E+n3 - 00IF-I
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2 7 4 +0 I
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
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1515
30
L2 hU = f+ 2h (35)
and T2h is the differentia- truncation error of the 2h
grid The comparison between (35) and (33) that arises
naturally during the FAS Multigrid Cycle without additonal
investment of computational work shows the remarkable
differential aspect of the multigrid method This is in
sharp contrast with other methods where the algebraic
aspect is usually the most important and central one
These properities open new and interesting possibilities
in implementing the FAS algorihm z2h in (33) satisfiesh
the following
L (36)
and the truncation errors can be represented iby Tkz kkUdL
(provided additional smoothing conditions) Then we have
cVk +()(7
Then we can change the FAS algorithm by defining the
following correction equation on a coarse grid (instead
of 33)
2k 3 (38)
ORIGINAL PAGE IS
OF poop QUALITY
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
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6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
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128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
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1249E-03
T170E-03 130E-03
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13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
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1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCEFF 558 ACCEFF 557
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
70IE+ol PNOE-o
7AI -OU
3
2
lo 9)4
303ia
+10
6137E-llO
IP~E-nlshy
thE+1)O
659L400
qalE+O01
SakE-00
I5 E-04
3
3 3shy
31105
R10 9a 3 1 L
17J+00
jF4(MO SAIE-nI
13A E+Wl
ahF41 shy- I(S -ol+0
oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
E 0
6IE II
pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
274k+AOl 57 E-ntl
qT -Fo1
2E-nl 234h4-00 i 1 E+10
VE-02 SPSF-03
01
2 2
z rn~1t 1 3732o shy 37 I4
37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
- - PA7E-O 3 7E-0
pqPE-0)5 63PE- 3QOE--
02f-f5
-shy
- --- ---
3 3
37114 7U3)
3ql -n5Eshy$nFtp IIA-tl
- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
-4u7 - 6 E-ot 03070 2op-ol
-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
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-- - - - -- - - - Z az tI-Z3zz
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C 7U3EO2b-R -67t723 6- - - shy
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AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
31
A series of numerical experiments were performed to
check these interesting points including printouts of
and I U -ii(we chose for this purpose problems where
the exact differential solution u is known) As a result
of these experiments we can point out the following
conclusions
a) The assumption that after only one multigrid cycle
(5-6 work units) we get the needed accuracy in the
solution (truncation error accuracy) was confirmed
In addition truncation error behaves clearly like
O(h 2 )
b) We can get the same desired accuracy without even
performing a complete multigrid cycle ie we can
stop the process before we interpolate back to finer
grids The solution is then defined on coarse grid
points only but the accuracy is the finest grid
accuracy This feature can be very helpful in
the development of multigrid methods that use small
amount of storage much smaller than the number of
unknowns in the finest grid
c) Using thd T-extrapolation (38 instead of 33) we can
get a much better approximation to the solution of
the differential equation on the finest grid much
better in fact than the approximation achieved by
ot 06
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
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3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
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556E-03 - 109E-02
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6316 7316 8316
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3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
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415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
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3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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31105
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17CF-04
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37361 3717 373Q5 37398 -
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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sp5E-oll 7QE-02
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UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
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-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
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- -
e wEND CYCLE NO
-
-
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----- -- - ------shy
- -
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
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0 b
1
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]t-- --- ----0-shy0 0 0
-2fshy
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
32
the exact solution of the difference equations
in this grid where the accuracy is bounded by
truncation errors In order to get this better
approximation we do not have to invest any addishy
tional work Our numerical results do not show
however that the approximation is 0(h4) in
spite of (37) This is because of the fact that
at least 5-th order interpolation is needed for
this purpose
Nevertheless the improvement we can get by this
small change in the FAS algorithm is impressive
The improvement will generally depend on the
smoothness of the solution The more smoothness
the better approximations
In table 2 we cancompare the results we get in the
different variations we tried The problem is Au = f
the solution is known given by u = sin (3x+3y) (04x43 Osy$2)
Table 2 LII i- tL
Size of the Experiment Experiment Experiment ExperimentFinest Grid 1 2 3 4
9 ) 16 x 24 14x10 2 (23) l8xl0 2 (59) 53x10 - 3(5 23x10 3 (91)
32 x 48 34xi0 (23) 42x0 -3 (56) 47xi0 -4(55) 23x0 -4 (89)
64 x 96 86x10 4 (23) l0xl0 3 (54) 52x105 (54) 18x0-5 (89)
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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ACCEFF 558 ACCEFF 557
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OUTPUT NO 2 Yd
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
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- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
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PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
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37361 3717 373Q5 37398 -
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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433 O5 357n I3n33
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- QOEshy2SEshy 1 49J(E-02
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3 a3 at)1I 13tl
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-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
ORIGINAL PAGE IS OF POOR QUALITY 33
Key to table 2
Experiment 1 shy is the exact solution of the difference
equations (up to randoff errors)
Experiment 2 - U is the approximation after only one cycle
in all grids
Experiment 3 - L is the approximation after only one cycle
in all grids but using (38) instead of (33)
Experiment 4 - Like experiment 3 but the number of relaxation
in each level was doubled
The numbers in parenthesis represent the number of work
units which are equivalent to the total number of relaxations
on the finest grid (printed in each line of the table)
52 Distributed Relaxation
One of the important application of the method of distributed
relaxations is in finding relaxation schemes with good smoothing
properties This is generally hard for very asymmetric operators
In these cases the usual Gauss-Seidel relaxation may be very good
or very bad depending on the direction in which the relaxation
is performed The problem is even more complicated in case of
non-linear difference operators because in this case we may need
one direction for some parts of the domain and an opposite
direction for other parts In these cases we can perform
as we explained in a previous chapter a series of altershy
nating direction relaxation sweeps but this cannot generally
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
- -shy
- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
__
OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
- 400- FqFutAT5 DELAY[ 1IEAU IqTIEr SChEE qES 4EIGTIrm EVENYiiHEPE -G [Oj)IS K- u4ERVLSACOASLST III 2 Y-1 -TEDVAL5 J 5tj
E_ TA IFLTamp i OLc _ I l Elh-
2LT9P1SANA _ET 1011 211)
EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -
cc _ -- - ioiM amp4 AEFS$ M shyf XoR
LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
F~J shy-- 1F4 J
_ S - no - IIFI - l7Ftn I3amphl LI CYCLE l
- I3F ---ln -I -shy - C E-l------ -IE-I S 251 - 0 W7AJ7lI IP3FfaIl OQE+III A14pF4II
- a_ - b __13A f I I1 I I - - S ECI SIU l11 __ __ _ _ _ _ ___ __ __ __ _ __ _ __ __ _ 51 -1 I Fl +111 +1pl E i ~ 14I Fflll~t
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
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9~ -it-0O5F rta
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- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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3 3 A+tIr-f77E-7
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
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183E-01 aonE-02
I 3
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I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
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I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
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q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
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0 b
1
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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AQ73 15A P 34Etl P F -11 365Enr -
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
34
be an ideal solution If we think for example of a threeshy
3dimensional problem we need there at least 2 =8 relaxations
O a multigrid iterative cycle indrder to doOn eampch level for
relaxations in all possible directions This may be too much
because we know that in most multigrid procedures we use
only 2-3 relaxations for achieving optimal multigrid efficiency
So alternating rirection relaxations may in some cases reduce
considerably the efficiency and in addition it will cause
more programming difficulties
Because of this and other reasons it is convenient to
find relaxations with good smoothing properties that are
to some extent independent on the direction of the relaxation
so we can relax always in the same direction no matter how
the operator behaves
As a first example we consider the one-dimensional
operator represented by Ei I - 7- Iti C C
which can be derived for example from the differential
equation 5-R1 (by taking one-sided first differences)
The smoothing factor can be easily calculated in this case
and for extreme values of it behaves in the following way
1 Relaxation from left to right
(39)
Relaxation from right to left
for
In the present work we check a special principle for
the construction of distributed relaxations The relaxation
4
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
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OUTPUT NO 2 Yd
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
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END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
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5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
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3 a3 at)1I 13tl
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I 3
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I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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---
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---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
35
is always-done from left to right according to the scheme
represented in figure 6
Figure 6
Relaxation
LX
and in figure 6 represent the values of the
approximation before relaxing on point i and the new values
after the relaxation on this point are represented by k
J is chosen so that the residual at point i is
zero The changes performed at points i and i-i make the
residuals at the points i-i and i-2 nonzero although they
were previously zero when we relaxed these points So we
can choose the parameter lt so that the deterioration of
these residuals is minimal say in the L2 norm
If we chose A in this way it can be shown that the
relaxation scheme is always stable and satisfies tV
for all 0 For P7 0 The optimum parameter
t depends of course on but satisfies 4l iamp 5
and in fact we can chose a constant suboptimal m say
= 45 without significant lose of efficiency
It must be pointed out that this scheme is not
necessarily the best one for all values of M In fact
there exists better stable schemes (Brandt7)for large
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
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- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
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377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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OUTPUT NO 2 Yd
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
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END CYCLE NO tshy
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
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-
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 3)907 I ft)I
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1 nno 34 t 3R50 3-657
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1 2E-nlI -shy
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-
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h 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
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3 a3 at)1I 13tl
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-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
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------shy
----
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I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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---
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---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
36
Using the same principles just stated we calculated
distributed relaxation schemes for general five points
operators in the form
d
a -s c
b
where abcd O and a+b+c+d=s
Operators of this type are typical for instance
in Navier-Stokes problem The relaxation scheme is
described in figure 7
Figure 7
Direction of the relaxation
oL and fare the solutions of an appropriate optmization
problem As in the one-dimensional case A+ A+-to JA+fA7
represent the new values after the relaxation of the central
point of figure 7
The results (calculated by the algorithm described in
Chapter 3) show that the smoothing factor is bounded far
below 1 even in the cases of great asymmetry of the operator
except in the cases where the problem degenerate practically
to a one-dimensional problem (like the case a- cz 0)
In the case of Poissons equation (a=b=c=d=l)4 =315 which
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
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ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
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P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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-- I7E+n3 - 00IF-I
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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
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3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
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18(F-() ns powENO CYCLE NO
5 b
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4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
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50 1 PS3E -( I t
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a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
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14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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-
-
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
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I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
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---- -C -0 c0
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------- -~~- -0- - -- -5- - N
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0 b
1
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-2fshy
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
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II 5E-fl2 - oqiAE-OM 279E05
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
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- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
ORIGINAL PAGE IS OF POOR QUALITY
37
is significantly better than in the normal Gauss-Seidel
relaxation (1=5) For this Poisson operation we get ri
For other nonsymmetrical operators the comparison between
this distributed relaxation and the usual Gauss-Seidel is
even more remarkable as we can see on table 3
Table 3
Operator Smoothing Factor 4 a b c d Gauss-Seidel Distributed
Relaxations Relaxations
5 5 631 J 313
105 5 105 5 913 449
5 1005 1003 5 990 481
5 10005 10005 5 999 484
5 5 10005 10005 1000 752
Like in the one-dimensional case it may be possible to find
more efficient scheme for the extremely asymmetric cases But
the present methods assures a good smoothing factor in all the
cases bounded far below 1 and consequently an efficient multishy
grid algorithm Moreover if the problem is nonlinear we do
not have to know in advance the behavior of the solution in
order to define an appropriate relaxation scheme In addition
This distributed relaxation scheme for Poissons equationwas implemented in a multigrid computer program and we getperfect agreement with the theoretical smoothing factor
= 315
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
- -shy
- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
__
OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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E_ TA IFLTamp i OLc _ I l Elh-
2LT9P1SANA _ET 1011 211)
EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -
cc _ -- - ioiM amp4 AEFS$ M shyf XoR
LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
F~J shy-- 1F4 J
_ S - no - IIFI - l7Ftn I3amphl LI CYCLE l
- I3F ---ln -I -shy - C E-l------ -IE-I S 251 - 0 W7AJ7lI IP3FfaIl OQE+III A14pF4II
- a_ - b __13A f I I1 I I - - S ECI SIU l11 __ __ _ _ _ _ ___ __ __ __ _ __ _ __ __ _ 51 -1 I Fl +111 +1pl E i ~ 14I Fflll~t
__ - h ) flq 4-un 1VE1P- EIIEtii
I llI I 2 dF4 tI 90E4 011 - qE I shy
-shy hA -- E -+-IliuI J - L2hS$ l3 il II7E+II - _ - 766 sr-ol -- lidB -lI 1)El5pq -- - +r shy
- -E -I A 1 F - - A I f 1 - - - - - -n - shy
- to I Ef hQ Q $OE (I -- - shy
--shy t( Pl3( FQ FIII 110 02F
15_ 1 IVnE4+110 I EIf I -n I E4 qli5 3 IE4 l IIEI R_E~f l FSi Ell 110A ktE4Ii
l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
U-
-~ ~~ ~A~7 F +Icw7c A - 11n-F+ PiI)hf~ Izzz-----------nshy -shy - -shy -shy -
a 5 9tr 9
- ___
A0 Eshy 1o) 7E-nl 4J IE O1
5F-1 t 7 E-)IAIFP- Il
77SF-l _
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It7 + -II -
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--
Q shy- - tql_ |rfl-oI
L| QL tE-I 2 -
9qt
P___hOhf001 (IMEt 3 I) 11P -
2
- - - )0 F-112 3I F-O I ) 3r0- - olIJ~f)I
- thE -- IE-O 331)E-13
11rE-O -n 7 0 -03 _shy -fl -shy SR-q
A711E IP I $RE-III
- 17F--lj - 1171F1(p
- 7pE-0 I I1shy-f-Qi7AF-iS
-shy F lP 77P - I
-
-
-
--shy
-
-
--shy
-shy
-
-
_
S- 5 ___43 4 3Aq 9 P97df
-
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-I-I - IIEn) 9SE-flI j3- En I7J -11I- -
454(t -shy -
SE-dnl-shy Fl-F I
- -shy ------
-FNn CYCLE MII
_ 5 R _ __--shy SnwFI __ 4shy A 1n11 PIE-1 t -
Al71E l P( __ AP -I tm I
P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3
___3 qK12ff1 YSE- _shy
PE-Ol 1s7E-nRhPE --) 9QE-O 61 f - TtIi - )
p
6-1 E-02 7PEshy
p I - Q7r -I
tE- I3 l F-iVI it I PE I7P0Ia -shy E3
9rn 1-)2 IAF-F-)
luV o-n IphEI I
tbcoF-112
- r
1 1 -I _ M -
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-
-
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--shy
-shy
-
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I _17 I 1 x - 110Fshyf _SEO 6LF II FI uI --shy
____ _ _I (shyMi3(l
I97(I-OAA -I
- tEi3 77fr~hRV
- -shy lI7PE-l t2
o -7It-I
PcF-n
S7~II_f3 u-11
Q
1117E-02a
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_ 1
A15
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CtE
AP
77 E-01 tl I Zll
74I-O1
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310-o
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QAF (3p HF b
1~
71 7 - P - 3E n 7-
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A
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123il
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14 -13-
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~Q3SI)11
-
7711E-c3 1qfl-n VO-03p
-
-
j4-EM -
l-_h 13 31 __
1033 V-IO-flP
aou1F-oe - 1 F-OAj 3MlF-ii
II~Q14PEeB2
0Et1
-
__
5 __
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it_ --shy
1-
~ shy4 3S
1I 11~75 431
3 uS- 14
-
-shy
9hlE-13 AhfF-v3 1AfE-n3
(E O G5E-03
b11 q)F6113
-
-
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t 77E-03 3POE-11
P1-PE-flIS 1 71I1E-05 01 6EO04
-A1
-
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4AE-i3
5U7Ed-3
3$AEE-a
P65E-113frII
1114E-03 At~IIE-fI4
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1 7P)F-1)
Ij
R-1 I bh-f-3
--II~~~~~j _ J$0 7t
IAQ 32~lnstzr--__1t--- -9
11197- 2$AE- Afl__50i ~oq __e113too - 33r-0i-----j1___ 35tE-5 $7A -115 1Q4JE-115 531fl-n5l
_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
E-14 l3-d i - IM3A II - 19 fqJ-Cu6 - lbI-
1 IJ bbl E-U t U E -I -
A I93nAP IP 3E-1 94)IF-011 I72E-03 -_P jF-n - _n3 fl 12IF-(3 QF -i I I ftAE6 - I P AF-03 ----
------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy
-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
-06 shy
S- F- 5 1E-I5 - RQ1U-s 4AI
- q7 oR-o p lE-p 1W 1 - --- shy-hq2E-n5 JlE-IoS
9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
__~ ~A P3 30) iflOE-rll I nIIE-All - AEf5ARE41S -- FN CY NJ) shy ~~ ~ I ___--- I)5IpE-nfj I 01-tS|ltF114 IIhF-tIII
____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
Ul
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
_ I -bS4 -
2ho17 933
s~~lq l|dF-iiS u1E53
- 71E-IA - 4t-E-16
E-(Ih -
III1 IF-415 7 4 (F-)6_
33E-16
PPF-1 I- r-n5
h 87Fshy 3 t9R _P 7rE- I)h -uaIP- Io- --2F--hb F - - - - - - - - - --- 2 __ PbR I 4E-117 bt3E-A 7 3QJP -t07 1 I 0F-06h
_3 I
~2h ~b ~lQ~rI$7 ____
FAJVtUQP-1A -shy -7 u _ __ pound 6F-flb __
53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
I rE-4 ER -~
rSIF-117 IYF-l fliE-Ie
-
-
- shy
- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
___
___ _ 6 Pi PAORJ
3)-I7E-05 211 E-rIS_ -
-1j-1307V-ol
Ok~tF-I qshyE-hIIF-tI4
~ ~ -
-
E -
___ S - 7a - __ r -- shy---------shy - -shy shy
-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
--
-
3 __37_E 37E-2 3)6E-AA--------IO7 E - - 3 _ 845-_ 7U E-I)1t_ 1- 9E-0Q 7 I3 F-)7
____ ____
3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
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5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 3)907 I ft)I
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
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0b77k-0l 28 E-0I
-shy
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31APO 39070
IMSE+o l14I7p
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6 6
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h 6
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flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
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1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
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622Ff-)
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I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
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5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
38
as in the one-dimensional case it is not necessary to invest
extra work in the exact determination of the parameters and
They can be estimated in a suboptimal way without significant
loss of efficiency
It must be pointed out that in the case of the Poisson
equation the Gauss-Seidel relaxation can be still considered
a little more efficient if we take into account the number of
arithmetic operations needed for each relaxation method On
the other hand for all other operators even if they are
small perturbations of the Laplace operator the proposed
distributed relaxations are much more efficient than the
Gauss-Seidel re1axation
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
- -shy
- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
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OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
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--shy t( Pl3( FQ FIII 110 02F
15_ 1 IVnE4+110 I EIf I -n I E4 qli5 3 IE4 l IIEI R_E~f l FSi Ell 110A ktE4Ii
l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -
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_ 5 R _ __--shy SnwFI __ 4shy A 1n11 PIE-1 t -
Al71E l P( __ AP -I tm I
P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3
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1I 11~75 431
3 uS- 14
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9hlE-13 AhfF-v3 1AfE-n3
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P1-PE-flIS 1 71I1E-05 01 6EO04
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1114E-03 At~IIE-fI4
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_ 2 pound_JoI _ 17VE-QI 4tlE- g - IP5E-1O _ 3 ___ 1 140 -7 523E-114 -- - 4 l rSE-I 4711p1E-04 h6F-nf
_ 15160 1E3 -I3E-3 _ --- 151 E-11 3 ld2E-03 5 3 I 322E-A3 21 GE-03 3AE-113 3oE-n3
I__ I) _ 7SE-n - 7ubE-I3 shy7111E-113 111E- _ _ -- I731 1177E-n3 - 21 E-13 3h 1 025F-13
----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
- _ -_ ] 777-a - 51E- t sIF-Ilu 17 IF-0S 11 17 9$75 Q E tE - 37 0E 0E- -- --O 1
3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
74 _ 3- 7 E- 15 I U15E- SI AI F-n r - N5IF-IIb3 I77 q q I i I UEIlU I PAFIll - I59 -EIII
W39 I - _ I s 3SF - d Jp-nj
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
--- I _ )fn91 A 3hF 0c 44 E-nS 7 173F-ilr 11F-I)
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
-- P1o 60 - 17F-nh17PFE-h degI 3F-A IhAF-Oh
1I 2 Q9E-QS _hIEQ5 2- plusmn) P7E-iPt
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26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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2ho17 933
s~~lq l|dF-iiS u1E53
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PPF-1 I- r-n5
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53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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- 784d _ qa iE-ii - -shy i6I4E-I - nbhFnA I PhF-I9
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3)-I7E-05 211 E-rIS_ -
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
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15deg90 2 1
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32E-07 1 t -7-shy306E-OA
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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
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--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
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2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
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3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
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3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
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3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
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3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
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18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
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3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
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5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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Ln
-
- - -
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- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
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------- -~~- -0- - -- -5- - N
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1
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--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
39
REFERENCES
1 A Brandt Multi-Level Adaptive Techniques (MLAT) forFast Numerical Solution to Boundary-Value ProblemsProc 3rd International Conference on NumericalMethods in Fluid Mechanics (Paris 1972) LectureNotes in Physics Vol 18 Springer-Verlag Berlinand New York 1973 pp 82-89
2 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Research Report RC-6159 IBM Thomas JWatson Research Center NY 1976
3 A Brandt Multi-Level Adaptive Solutions to Boundary-Value Problems Mathematics of Computation 13 1977
4 Y Shiftan Multigrid Method for Solving EllipticDifference Equations M Sc Thesis Weizmann Instituteof Science Rehovot Israel 1972 (Hebrew)
5 H Lomax E Martin Fast Direct Numerical Solution ofthe Nonhomogeneous Cauchy-Riemann Equations Journal ofComputational Physics 15 1974
6 M Ghil and R Balgovind A Fast Cauchy-Riemann Solverwith Nonlinear Applications Research Report CourantInstitute of Mathematical Sciences New York University1978
ORIGINAL PAGE IQ OF POOR QtALIT
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
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4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
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160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
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- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
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128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
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1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
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1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
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o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
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2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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3lhF01
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659L400
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31105
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17CF-04
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a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
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01
2 2
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37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
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37taA 37 555
71sE 0 Isk+0A
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H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
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-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
APPENDIX
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
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1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
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ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
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OUTPUT NO 2 Yd
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
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3lhF01
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659L400
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3 3shy
31105
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17J+00
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1023 065F01O
17CF-04
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a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
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01
2 2
z rn~1t 1 3732o shy 37 I4
37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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3 3
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
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37taA 37 555
71sE 0 Isk+0A
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H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
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--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
-4u7 - 6 E-ot 03070 2op-ol
-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
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-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
OUTPUT NO 1
CAUCHY R EHAN P2) UX6VYCSo(I) UY-VXFINJECTION SwO F=XD24Y2 StrMAtG gELx ATtOzf COARSEST GRID I5 - 2 X-INTERVALStANO 2 Y-INTERVALS HO 500 NUNRER OF GRIDS 6 LtvEf 6
METHOD PARAMETERS ETAu 600 DELTAS 200 ERWi 5000 - - TOLm 1O OE-04
LEVEL RESNORN(I) - RESNORM(I ) WelGIlre tsWdo WORK (c)0 --shy -shy
6 4T3E400 311E-i2 oTSE-01 1000 - _ - - - -__-shy6 449Eo0 A1TE-12 T48E-Ol 2000 - - - - - - - - shy5 413E00 460E-12 688E-011 22505 377E00 S17E-12 628E-01 2500 4 3E E600 493E-12 - 534E-01 2563 4 9266E00 496E-12 443E-01 26253 193E600 9432E-12 322E-01 26413 134Eoo 409E-12 224E-01 2656 shy
z 9806E-01 331IE-12 l34E-0 2660 2 464E-O1 29TE-12 7T3E-02 2664 -2 4264E-01 270E-12 441E-02 2668 -
3 280E-01 6952E-02 126E-01 2684 shy0 554E-02 2699
4 36TE-01 -15sE-01 1192E-01 2762 3 183E-01 299E-02
S206E-01 592E-02 836E-02 2R24-3074 5 456E-01 1OE-Ol 2QlE-0 1
5 6264E-01 - 727E02-I1OSE-01 3324 shy6 488E-01 124E01 184E-01 4324
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
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CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
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ACCEFF 558 ACCEFF 557
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OUTPUT NO 2 Yd
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
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5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
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A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 3)907 I ft)I
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3 3shy
31105
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a a
31 (1660 31 12 AhlF+011
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01
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37328
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shy I62EA I
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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
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0F-a06 lQOt-051-0E -2E0
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3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
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3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
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biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
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shy
r 5
31APO 39070
IMSE+o l14I7p
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6 6
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h 6
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--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
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I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
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PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
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---
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---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
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- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
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0 b
1
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-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
6 316E-1 -shy 784E--shy 118E-01 - 5324 5 242E-01 747E02 103E-01 5s74shy 5 187E-01 729E-02 919E-02 - 5824 4 148E-01_ 661E-02 -shy - 797E-02 587 4 116E-01 - 608E-02 - 99E-02 - - 5949 3 852E-02 459E-02 525E-02 5965 -
3 9608E-02 343E-02 387E-02 5980 2 - 343E-02 IS9E-02 190E-02 - 59 - -shy2 3
ol40E-02 1OOE02
556E-03 - 109E-02
696E-03 1OTE-02
5988 6004--shy - -
shy
4 195E-02 145E02 153E-02 6066 5 6 6 -
395E-02 775E-02 52TE-02 -
__
160E-02 -187E-02 i OE-02
- -199E-02 285E-02 204E-02
6316 7316 8316
5 391E-02 125E02 170E-02 8566 - 5 -----shy 276 E-02 shy - - 119E02 _ 145E-02 - e16 4 20AE-02 - O1E-02 118E-02 8879 4 - 144E-02 9932E-03 - m102E-02 8941 -
3 -
3-764E-03 107E-02
-622E03
715E-03 773E-03 646E-03 -
8957shy 973---------
-shy--shy - - ----shy2 2
- 502E-03 9276E-03
415E-03 307E03
j429E-03 302E-03
8977 _8 PO
- ---Oshyl
1 - 949E-04 o-10-5-E-03 - d103E-03 8981 ---
I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -
3 -12CE-03 o197E-03 il84E-03 9002 4 - 297E-03 247E-03 255-03 990-64 - 5 686E03 - 289E-03 355E-03 -shy9311 __-shy
6 9139E-02 - 320E-03 498E-03 10 314 --shy -6 - - - 920E-03 207E-03 325E-03 - - 11314--- 5 696E-03 174E-03 261E-03 11 5 502E-03 158E-03 216E-03 11814 4 4
-
377E-03 267E-03
128E-03 E108E03135E-03
170E-03 - - 1187T 11939shy
3 190E-03 -745E-04 shy 937E-04 - 1 955 shy3 q124E-03 510E-04 633E-04 1971 2 76SE-04 - 279E-04 -361E-04 11975 2 423E-04 156E-04 200E-04 1979 2 257E-04 - 105E-04 9130E-04 11982 1 -751E-05 1 - Al42E-13 _
-358E-05
- 108E-12 424E-05 927E-13
1983-_ 11984 -shy - - - --
2 651E-05 -444E-05shy - o479E-05 11988 3 20E200-04 181E-04 a191E-04 12004 4 613E-04 331E-04 378E-04 12066-------- 5 -129E-03 ---- 443E-04 - 585E-04 - 12316shy 6 6 5
1249E-03
T170E-03 130E-03
_ o509E-04
- 320E-04 0270E-04
1839E-04 550E-04 442E-04
13316 14316 14566
5 940E-04 249E004 9364E-04 34816
4 4 3 3 2 2 1 I142E-13 2 3 414E-05
4 5 6 6
698E-04 479E-04 329E-04 203E-04 s 126E-04
0676E-05 296E05
183E-05
sI12E-04 243E-04 463E-04 315E-04
o2l4E-04 198E-04 156E-04 i136E-04 o876E05 598E-05 o173E-05
1181OE-12 230E-05 424E-05 601E-05 9742E 05 832E-05 556E-05
295E-04 244E-04 9185E-04 147E-04 9940E-05 611E-05 o194E-05
_ 100E-12 222E-05
-422E-05 o687E-05 102E-04 147E-04 988E-05
CYCLE NO 2 CYCLE EFF - - CYCLE NO 3 - CYCLE EFF
CYCLE NO 4 CYCLE EFF CYCLE NO 5 CYCLE EFF
rog AcLt
14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93
CV(FcentrActOI
-
-
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- - -
ACCoEFF 574- ACCEFFamp565
ACCEFF 558 ACCEFF 557
2 l- - --- - shy
__
OUTPUT NO 2 Yd
jE irjnJs - _ srn s 1t 3 (2 DEL(V)ZPYG (t)--Y lit1rI~II V~ F- 43Y) G $
IjIM111 VAU S- y +y(1y23) PC)
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E_ TA IFLTamp i OLc _ I l Elh-
2LT9P1SANA _ET 1011 211)
EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -
cc _ -- - ioiM amp4 AEFS$ M shyf XoR
LFVFI W(I-lt (WE IC$TEDI 41 VL 1t E 3 -shy l e l~l 1 51F 0 1 - I Ij flj p l1 il+ 1 2 E+ 11(
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
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3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
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183E-01 aonE-02
I 3
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I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
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OUTPUT NO 4IO LAO
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
(l - - A 11 - PIIF tIQ I 7P-11 a IqUE-Ild 11P E- Itl - --I 9) 7 __I_ r- - Q thE-0s5 IF-)I 71F- i -
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-- __ PiIUAS I| tE( bull 7F-nS _ 07E-(9 -IE-()
__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -
-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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S- F- 5 1E-I5 - RQ1U-s 4AI
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9 3Ii(l V(IU II I6IF- lJ - - -1PPF-nq -oa P2 14APF1114 - - - F 114 oc(j5F (l
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____ P1-55 __IEil tlE-05 Ir-al -nl ---- -shy
_ S 23$1) IIOF-o0 l fE-i$ 77E- lS - P 1F-O0I _ ___ 2 bl 54e-li -f 302F-Ob 0 JtnFIl5 7TSF-ilc -
- L -45_ 1i17F-05 P3E-nS 3 I F-9 kP7F-n shy
_ 21odI 76E-n5 1 lIE-O5 ISPE-oS PM I F-fl5 shy
3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
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9~ -it-0O5F rta
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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
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359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
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PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
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2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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3 3)907 I ft)I
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U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
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6 6
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18(F-() ns powENO CYCLE NO
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433 O5 357n I3n33
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sp5E-oll 7QE-02
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3 a3 at)1I 13tl
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I 3
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A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
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7AI -OU
3
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659L400
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3 3shy
31105
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17J+00
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13A E+Wl
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oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
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shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
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02f-f5
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- --- ---
3 3
37114 7U3)
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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
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413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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-- - - - - -
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---
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----
---
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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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C 7U3EO2b-R -67t723 6- - - shy
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II 5E-fl2 - oqiAE-OM 279E05
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
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----- -_---- ENO CYCLE O6 S__ 7 67Q I EAE-0l3 10 1F-(03 - th6E-13 l 1 -IIi3 -II IA_ __ PIQ(E- 1 _ l l h~i -ll I )AE-13 t q)E l - shy
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3 17QS 33PE- - INIF-14 $ PQAE-(0 50E-l0J --- 1 17 1)A IlE-0I - 11IE-IlII t lIF I)J 39 F-1IIshy
_ I 7u7fl 5eQE-15 372E-05 7hE-S qQ7E-I -shy
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------- -- - - - -- ---------- -- - -- ---------------------------- CYCLE Lf _ - 20( 2i II-_ -- 5SE-( i - 4 9I-01 1 IjIIL-il -
S__ 21 n0 - - 150 EfJ -) - nf -im - -shy- 31o~~l - gp F-( shy
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-- 2flAP _ 1t -b0h 133E-n6 IIE-ob t JE-07 - P - I q7) M hEIN I P3F-IS - 6Q -0ASF VF-l 116
P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11
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3 PPQsh II4 -centA 7PE-I~b I021-15 I 7A- - -_ ___ Pt hn 37 lE-ilh puF-Ot 377E-)h - WAUE-06
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5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
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_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
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SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
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OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
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FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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2 7 4 +0 I
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6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
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131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
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3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
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3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
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4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
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OUTPUT NO 4IO LAO
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
--
-
3 __37_E 37E-2 3)6E-AA--------IO7 E - - 3 _ 845-_ 7U E-I)1t_ 1- 9E-0Q 7 I3 F-)7
____ ____
3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
70IE+ol PNOE-o
7AI -OU
3
2
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303ia
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6137E-llO
IP~E-nlshy
thE+1)O
659L400
qalE+O01
SakE-00
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3
3 3shy
31105
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17J+00
jF4(MO SAIE-nI
13A E+Wl
ahF41 shy- I(S -ol+0
oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
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pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
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01
2 2
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37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
- - PA7E-O 3 7E-0
pqPE-0)5 63PE- 3QOE--
02f-f5
-shy
- --- ---
3 3
37114 7U3)
3ql -n5Eshy$nFtp IIA-tl
- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
E-i -
-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
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----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
26uP p RIE-(I -F-15 I II -oi 11 7F-1141 )_- ------- - --------- -- - - - ------- ------------- CYCLE M- - shyENDshy
5 a 2 I F-Ob - S--I)q - - YtCLE P9FOF _ _P - A 17__ c55 -L I I Fq-i-S 7E-05 5 shy7shy 1
-- -I
-
_ I -bS4 -
2ho17 933
s~~lq l|dF-iiS u1E53
- 71E-IA - 4t-E-16
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33E-16
PPF-1 I- r-n5
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_3 I
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FAJVtUQP-1A -shy -7 u _ __ pound 6F-flb __
53-f30E~-tp7 EI h-S 6 - b --S _ n -
+
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3)-I7E-05 211 E-rIS_ -
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Ok~tF-I qshyE-hIIF-tI4
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-- c - 24A1J7 3 17E-16____ 171 E-oIb pbRE-l6 5IUF-0t - - __ __I _ _ 16E-16 QF-06 iA ___PSOE-p - 1 1 F-0A6 ---- 3 P S __ lF-b _ 63E-nl 0 F-0l7 1 7tE-1
_ 29 0111 71 t E-17 - IItE-O7 h Eh-07 I17-F-00shy 2 __QUI Fh-17 I7-3F-07 I I1UE-to7 - 1 1E-
_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -
3 9 1 1 IE- 1lE-l) l I 4E-7 71IF-t1l 0 t07_102 I- h -- 5E -17 IP+- -- It7F-OAI
- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h
_ 6_ 1 al7 Q-I 3OE-Oh I 5 I 5 Il rfF- )7Fshy---- p I rlmS~F0Ib 1 orr-n-Th 05
9~ -it-0O5F rta
- I Ad -ih lME-6 75-nh--------------- 13E-dt 1shy- _ _ _ 3 776 1S97E 0 A E-n7 - I IIE- 6 3l tl--
- 7 3 -) - ( - 9 -- jI~ 7~4o~il - - 11P I r-117 ( 1371j-ilt--------shyM7E1)7 I
1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_
2 )Il 23 31
3 3 A+tIr-f77E-7
_h bEI) A y - 37 E-n7~tI7gE-1l7
471E-118 aA(E717rE-+77cE-07 50O -ON0shy
1F- sr-n9E
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3301 l$4 -Ee7 31 Q 5 7E 7 -
15deg90 2 1
jqcF-f7 7 3E- 117
IC_ 6 l b1
32E-07 1 t -7-shy306E-OA
F-17 f711 E- 7 2F 7F-0ks
-
- -
- - - - 35V19 -shy tAA-11H OcSE-O7 lb IF-Oh 2- - - - - - - - - - - - -shy
~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -
~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy
1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -
____ _shy 3$131 __ IqqF117 1 j ufF-f7 I6JE-07 3JRF-1i7 - - - -
359 thfE-u7 TbIE-PR 11 71)7l SF -0j 3 qq 17- R E- cB 2IEDampshy - - 4rIshyh 1 1dA I 1F9
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
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L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
2 3n J49 1 F4 0PIE-n nAEo IsinE noI hl7E05
3 AQst 1734 cent0o i1E+0i( 1 SE400 176 =O
3 3)907 I ft)I
35 F11-n jV3F+nl
|75FfEshyl 2u7F+Aj
3lhF01
70IE+ol PNOE-o
7AI -OU
3
2
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303ia
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6137E-llO
IP~E-nlshy
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659L400
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SakE-00
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3
3 3shy
31105
R10 9a 3 1 L
17J+00
jF4(MO SAIE-nI
13A E+Wl
ahF41 shy- I(S -ol+0
oISUE+O(
1023 065F01O
17CF-04
a5F+AQ11 641E-05 -
a a
31 (1660 31 12 AhlF+011
I67E+o ISIE+n
762E+01
95UE+00 37nE-0I 4O O
4 3
31 0141-O -1 317070 5750 -ni jq
E 0
6IE II
pj]lon161)F4041
7021 -01 SM -flil
5 a
3Z$092 39 1)s
274k+AOl 57 E-ntl
qT -Fo1
2E-nl 234h4-00 i 1 E+10
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01
2 2
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37328
-5 u0--q 257E-C0-2tqFAl
shy I62EA I
76Eo 1ZIEo -hQTE-02 777E-01 - 31RE-01 -shy
SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1
777
-shy-
shy
0F-a06 lQOt-051-0E -2E0
-
-
- -
3 571Uda PLF+ 351L+Ofl 1Iliutho tl71 F04 3 113jq I d2~oO 6hE+II0 lqE+00 IJF-o 04 2 3 3 2 -
37361 3717 373Q5 37398 -
1IQ -I hUE-oi 2ubEriI 260E-0
-
2B31)E-IIOEkfo S7E-(
367F-(2
- - 6AU2EO IhSFOo
- - PA7E-O 3 7E-0
pqPE-0)5 63PE- 3QOE--
02f-f5
-shy
- --- ---
3 3
37114 7U3)
3ql -n5Eshy$nFtp IIA-tl
- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05
-
-
Ul u
37taA 37 555
71sE 0 Isk+0A
-) 11 7E401 2R15flO
H iMlt+oo IhAE+on
I IIE0113-- 705F0Ls--shy
- --shy
--
3 17570 -F) - 3 7l-nlIPPP -shy 2 E-01 - 15Enashy 14 17633 15lF00 P41A0of tIAbF+00 339F-na -
U 37A 59 I E-1 7 E-A-(-l 237k-ot PI F-04 a I7758 S II -07 137E-fl 11AE-O1 156E-0 C r 5 -shya -
1 nno 34 t 3R50 3-657
u2sr+o 3kn0 SQQLI
1 2E-nlI -shy
hUqE~noI uViE +10 I)L+0Oshy173F-11
-
-
biiE~fl 177E4nP
0b77k-0l 28 E-0I
-shy
A726i13 2t)E-03 htlt-03 QiAEtu
- -
--shy
shy
r 5
31APO 39070
IMSE+o l14I7p
-shy 1s68F+00 3A4E- 1
-shy 53Fo+00 I6hf-0l
- 1IAQE-03 SE-03
6 6
CU7CI41n4
1AqitIP03E+no
- 24AE4np I3E+ol -
21gt~E+02 237E+MO -
50260 93E-02 --shy --shy -shy
h 6
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-a
-shy I12E+0(0 57E-oI
--shy 72iE-01 293E 01
flOE-n2
18(F-() ns powENO CYCLE NO
5 b
433 O5 357n I3n33
- 21060-shytI E-1I
IflQEtll
- QOEshy2SEshy 1 49J(E-02
-shy-
sp5E-oll 7QE-02
262P-ll 1
h0A-3l nqbs-03
1111603 --- - - -
UI 41h4i5 - qA7E-OPshy - I SUE- I 1I- 7E-101 - 011 -fOillI - - --shy
3 d37 - 0 7UE-02 7-IIFn - - Q FIE-vt IME0shy3 2
437V
417 1A 121I Fnj
84hpnl
7 -t2
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-- 25 E-01
423-E-02 -
71 LF0 I IE-nS
--
3 3
P
a3i7t 1
413 7 60 - -
bVIE-0lI 37h2131l 1 PF1
A tIoEI PA 7F-o
Pshy1Et(I2 -
9AhE01 031F02 3276fl
lqE-05- 1 06E
-997E -olh-shy - -
-shy
_ ___
I 9178t 2hE-nl 41HE-01 - 320E-n 33E- - -------shy3 a5747 -SQUPOp IVE1-01 - 6172 nI8SF05 2
3 a3 at)1I 13tl
I5aF2 6b5E-fA
)S-l7PLIE)-O shy
PIAE-02
I I -01 AAOEnh 12116-Os
-shy
3 UL3832 L4 fiI 11i ASJEWn UatEshy -A 11 3E-09 2 aI33 I(I006-02 ISOE-02 i IF02 - 37Q6-flb - --
I 3
33 A 2 1387
U0hi-Ip 21qEnp
622Ff-)
048Eshy5U2EitP
- 236E-02 I41(T-A6723b-n6
------shy
----
-shy
41 4 3- IsqFE+11lA 90E+nl 2 iE+o0 2PIEwOU - - -
I S0I2 PAAE-l I q6PE-11 P12E1O I uF -04
3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
5 u3SA (IIRE fn( 137F+60 I22E+0n 377E-na 5 5i0lSOO 57F-(ll I A1E-nl - - 144E-2 agqEou shy6 IO0A - ZUIEA4i 341E402- -- 35k+1 I h3E-n3 - - shy
6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
V 5 - -52s E r A 1-JE _ M E i I6E 2 3 I 6I E-15217E E-I11 IIIE-AH E -fl5
A 31 _ 01E1t7bEQ - -P - _35_ I - F -- I _ 354L 5EtJE-07 b aA - 7 - E7-Oamp |E-I)L
L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4
37a4btI- 7S J 04dE-117 MIh E-97 9AAE-117 -shy
- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn
SCYCLE NO 2 CYCLE EFF 64i1 CCEFF )41 -
CYCLE 0 3 CYCLE EFF 524 ACCFFF 3 CYCLE NO I CYCLE ERF 5A5 ACCFFF b8M3-------
cyIE Nn 9 CYr E FFF aSA ArCCFF ShJs- CYCLF 14ih CYCLE FFr h 6 ampCC FF 591
CYCL E 1 7 CYCL E FFF 6 ) AC C F F S97 C YCIE N A CYCLE FF A36 ACCEFF 2 -shy
CYCLE Nrl q CYCI E FFF f35 ACCEFF 106 CYCLE hOlI rYCL7 E EFP 612 ACCFFF hII -
E wcaiEI CYCLE FF A51 ACCEFF ble q - --- CaLIE 121P -- Lu I ACC FFF Iq - shy --- - shy
--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE
Ott G i CL F CT o
o shy - - -
00
co
OUTPUT No 3
PtWMHWMwM 4 N iMWNW WM iHM d HH eMHV841 WHA MWe14 AM bebeb Wbe4)WM WAW 4W MAW 4vi MIr4$ 444W18MWIM AMSwJ W 1W0tiM f4 MwH 441g kMtM 1tH 41 gMWH H4hI i M
TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
-FI1 350 3I(Q - - - -- RFYNOLt) 11FR 41OE+ -shy
6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -
3P50 7 +06nE6 -wE m-n)so n28E+U o 1147E+405 3QEioS JME+bit-E+04
9E+n3 h+E+A3 - -E+03 6 9PSO 8iAMA dE+(
--oN) CYCLE NO 1130r4 qnats t YLE N shy 6 b25O I6AE403 5bE (13 lft+A3 5 m mm mm sgmmm m m m~~m (
-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
1hFsh dOAE+t3 ptiRE+o3 9RSE40167SA - IA3P IIOE+3 TjI3 2AIF+0shy
b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
u
3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
-- I7E+n3 - 00IF-I
2 q1 71QF+np = lh6E+ 2 6910 - 57ni411 -- 357E+02 shy- 9F+n3 shy
PQE(12qlq -Q1E+n2 - 17DE42127E+I 7qE+(3 ---- 70hF403 I46E-012 h919 - ShFO
90$ 156E403 IiAEo3 71QFmO42 P q17Fn2 - 2QE-n2
2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03
2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
1 wEo I A E P3 I115E+03 403E-013 6)67
3Fn F + OO11 7 1 17 g fl hF (13 6 7 E + n 3 I20 F)3
2 7 4 +0 I
s )P l h P7F + )2 r t+n3 - 6 9F centO I s qE + fI5 I 7oI oh lIOE+ 03 I5 iE~n (aISE + 3
5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni
6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0
2 12 A30 InE A2 I S E 2 I3 E +02 IlQE- na
3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01
I P 6 1 S 6 F R + 0 2 F + (3 UP SE +1)2 I (19E - 61I
2 1 P F 34UF401 11 PE+nl A~tE+nI P(F02
131F-402 23 E n 613AE+02 110TF-n23 1289h -
3 1 412 Sh tF~n I bOHE+n q93p+Ai 308E-n
11 1 75 A 2 F + 3 2 E+ o I P+ b ampO 9 7 2 6 E - 0 1 eshy
4 13037 Ll4QF~n ttII 276F+112 (E-At0 17 4t (02 p32E-nj -
S 13 tOOn1+pI71 2 E 02
3 1$ 1it anaE+Ol 6t7F+01 53I)E+01 5 7F-02 C
3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02
2 lt its5 t5 P4 t 3t E~n I 1 3sF+O1 I OPP-02-
3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (
3 131Ih6 OQQ F4nI Il7E+02 87qF+ I I oF -Op
133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63
3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO
3 1 P 513F+01 663E+nl 781E+01 09~F-63
3 13 p3b 2cqF+( I57qE~nI 2b EOt 615E-03
13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S
494E+041 Lq$E+O4 467E+O005 11611 S3u E+og 2H7E~n2 2q5E+00s 13Ahl i 7 F+n3 33F+03
i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
- - END CYCLE NO--Q 5 18361 1qEn2 7E 1 - 496E+02 SQqF+n0G5 l86it 1n0 +112 P35E 624E+0t - L26E+1O a 1d 67L 82OE+Ia hlSE+01 - 17PE+A2 t12E+no- - a tt 7 30 lq97F+n1 U+o2 902E+Ol 51IE-01 3- I75p - SuE~nl 320L~nl 9OQE-Oa -3QAE+01 U ip8iIu I~nEn3 2311E413 - 3311-3 3 P64F-01 - shy
- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
dI J3q lalF+n1i 2624 nj4I 41L5E+01 - qpu~En3 ---- - -- -
I LAss 17Qr1P4 73t)E+on t F+O6 CS5E 3 lI 5AnIA P2 F 1I 347F+ni Phigt4Ot 40I3E-n3 ---- -- shy
4I -5OA I(OF4oII 9+(- O I - I IIFo I - 3ASE-03 -shy ______shy
3 240O 6IF+n 2bi+E00 nqmE+GoCIQ1 +O0 a 2r158 147TE+01 ZhE+nI 1$5F+ll I - 313E-39 52bp RLF4A IU7E+nt bS+ 3jqFfl3
5 5J7I P31 -+o3 SCt+I3 -- 111IIE+03 139QE-0l5 25721 uc1ir+nI q3blE+fl 2AdE+oI lIJF-0I
5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
5 6f)21 2AUC+Ae 5~33EWn 71E+00 SpqEA- --- -- ------
IS 2721 7h IPttc13l IO1E+n4 -- I( +0U IBIE+00 b 29 12It 19 Ut+ A2 513E+np LnF IIEO -- -shyn 8F + - - shy
6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01
s 31) 721Si--S ~~ ~ ~
I7E i t-- pF(iEI -shy
-shy I E On p6tlE~ont-shy
-I IE I h47 E+oo
183E-01 aonE-02
I 3
3nw 3n677
I48F460 3AIF(10
I h +00 6037E+00
59F+00
65nF+O0 332E-03 173F-tl
-
I SOAAh t 17F+00 -- IIAAE+nO I 7E+00 W( E-
2 30890 5 (AF + nit 7 - t S3+I F l tAOE-O
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a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
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aUa hQs W E -P l iSOFt a - 3 7 9 E- 0 1 - QhPE - o - - shy
5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
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-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
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S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
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-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
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CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
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OUTPUT No 3
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TOKEq (1)(DEL-RUODX-RVOOY) U-PXFIEQUATIONS - NAVIEshy- () (EL-IJIO-PVA(OV)V A PYF2 -- (3) tLX+VYFI - F FUF2F30 -
VxSQRT(R)FXY PR22(X2+Y2) (FXYmI+X(I-w)Y(CI-Y)FUNCTION VALUES -- JYSQIRT()FXY8Utr)ARY CUNITIOHi AND rhtrAL
FASLIKF INJECTION3FPAPhTE RampLAXATIONSLINEAR INTMETOD
COAPSEST GRIm) 15 2 XINTFRVALSAN() 2 Y-INTERVALS Hil 51) -(IF IbS shy --(WUZER L EVEL 4 44 XG4 ampRL
METHOD PARAMETFWS ETA io OELTAu 300 T ElL s jnnEnu
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6w O 1J W tIVJ41 8 aE 3 i-LT-2LEVEL 0Rt - shyEo -shy
4 ~ 0nE09 kPARF41)36 1OnIoa 37utr+nm
END CYCLE NO tshy
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-- S 511lr03 QIAn2 136F$(ILI I E+1 - -
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b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O
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3 $153F+nl IRA E3 iAE+O0I58E+13 267E+03I 60O0b
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2 0~ o3E+024e~ -- 166F+02 Il7E-2S2
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2 7 4 +0 I
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5 iti 7n $1 +w p sqfkk +12 352E +02 10 1E+o
6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5
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6 t217 0 30 r 02 4 PE+ (1 297E+02 nek+0 1 = -END CYCLE NO 3
-0
5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t
5 1h7r 33QE+ti 6R4E Q P7PF+D 12 QE +6
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131F-402 23 E n 613AE+02 110TF-n23 1289h -
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13 qq_ 8 ujF 07 IibE+t3 125E+03 - I(+hE-n
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i+)It11 7FT +n IP3E+02 P2iIE+001 5+nlS 151tt jheE~n q Isq k 0 s 1 3E+05 35
A 171II t5En I7F+P 130E+nI AIE+0 I 6 18111 8A4n1 - 18 fLA2 - 62ampb1 tIOF+OI
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- I - EiA7 - abt6367F+01 b5E - -- - - 1dE3O7 11M439 Ptl)F+Al I A3jE~o1 PPf3F+0I 933FA ------ shy
141t19 a qbAP+fi I33Efl - - lLt+01114 I4E+oO --- -- shy
IUa39 -234K+02 tS2EoA2 - 7QIEernl10 J+I- 8+ 5 19 nd -- Ih E+1l - 3d+6 - I20e+0I 5113E-0I0
6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
6 2loAC 1IQE+)3 313k+03 - - 272E+02 SE A6 2bt 27tF4 I 7E+0l - -143E+O1 SF+00 4 23689 pIiQ 401 4L69r+ot 896Q~F+00 15E+0(l 3 a3 END 0+0 CYCLE NO- shy5 a3q- 17F+5l 7E+eFl1 IiE42 6lE+005 24 -1-927 F+III 4 1V4I -n 4E+O - IOqE+0)- shyu aj52 I7E 0I 137F+I0 3h1E +0j q 3ni --- I 243i3 231E nl f+ni - 1L4IE+01 I I -3E-0I
Fn46E 64 5E+(10E+0 21E-11 3 P133 4 0 shyd a39Q3 3qEn WE-02IiSE -616E+02
LI - 2ub -- 317F+Al 533E+o0 3 7 3 F+ I -- 3A - aSI0 a---- 1A3---l-2+ +3r031 P t --- 1 liELo - - -3
3 2a 5I+n0-Uq t +f 0 3 n SLI5Q6 it7+ I r35Eol 9 9 +1t - 145E-02 -
S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy
3 p - - 67L p AE o SIE+nO---- 30t1so+00396E-03 UI P4736 - 3 qSF4 nl II +nI - 7tF01 - 701F03 --- shy
176FO1 - 11 nP701 11- E+01 -- -- - 700F03 - I - 2d$IU +n k+ 0O 75F-03 ---- --- shy2) Fff aI) P3 3E0 -o - shy
a a t7 7 - 3PIlF +I11 1ta71+ UhL40+01 536f -03 - - - - - - - - - shy
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5 P597L 6qQO~F 6 -13SE+o I 640E4+00() 121E -0q I -shy
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6 2qppt 811F00 - I 606uii - 683E+00 - 117E+O00----- -- - --- - - - -shy
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5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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OUTPUT NO 4IO LAO
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6 PA~6RQ - 3OqE+n)4 - 42PE~ou -- Oq~bFtOL04 746E+00 ---- - - -- - - ---shy
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
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I-COZZZ 0000
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6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
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a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
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3 U4U60711 2a sJ 27E0 -IAA~ Lashy -
a UL2133 276E-1li a27E-n - - 3626-01 460E-05 - d atlaIQS -aa 5 7E-ai - 177E-2 - 2861-os
5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy
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5 44a945 I1UOF -01 233E-ot 16713-01 taqE-on--- --- -- --------shy
4 LEAnA IAF a-E-92 -- 0 1uE-O 2E-O0 -
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6 U7 5h8 I 6 + vIn Pq2E+oia 673E-fll - iP6E-03- ----shy
6 L850$8 16AE-11l dMA3501 1Q3E-01 l3SE-l3- - shy
6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
S U9Q758 -514E- IbF-aP- -pIE-0I 136E-03shyr - SO 006 P23F-oP MI~E-12 - 1S-02f) - QUF-0Li - - - -- - - - --- shy
I 5n070 25PE-I1P I23E-112 5SqQk-2 Pb6IE-flu l 50 133 2 0E-u 2UE-o12 221E-02 46E-04 - - - - shy
3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
3 50 J34 7AgFO1 153E-o2 -- 71 IF-03 I136E-Oh -L - J - ~ -- -- - shy
2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
3 rkl0 l2E-)3 33hF03 - 214LF-03 612E-074 5033 hcSE-11 bQ7E-aI 63[E-01 443E-05-shya S39 8- ABPn2 - Wl6E-02 2k52-05--shy14 1 U57 6 I hE12 161 F-as shy09F-03 104I4E-13
3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
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6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
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14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
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3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
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4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
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1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
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70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
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-
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OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
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I-COZZZ 0000
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CYCLE NO 1i
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5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
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6 o C418l 711 FoA) I PAE-nl - 7)PE-A 377E-n3 EN CYCLE No
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3 5otl 1PE-On E03E-nl173E-Da - 3014E-AS- - - shy1 51 Ih 23F-o2 3EOA2 1 SE-02 III E---shy2 6 E --- -- ------ ------------ - ------------- N ---- -shy2 - 28EE 3 - 1) AS - t31 111 01E -0i1 h 17F -I S5o IQ PStF- 53E-o -2 16E-0l2 32E-oh - 2 50(3 172E-03 IM8E-03 - 312E-03 hh6iE-n7 - 3 5ad9 AA-2 -319 -- 3 E0 ME-Oh --
50 1 PS3E -( I t
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2 S03~~ - hE - 63 1l70t-03 II~i ~~-7-shy3 - 0 -5 lIIPE -AP 141E-nA2 7 9 L0 P - M62E-07 --shy
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3 511 7 3 16E-0 21 - 7E-n flhE-06o -- - L3 0 - l ti AOEn b1b-J shy5n 35 L63E-na F n 2 - -63hD - - shy
a 5n5qm L UPFaI)I 7q()ujE-on3 U7Uh-03 513E-06shy55 1951000 291F400 5AL13 267F+001)Eo
5 - OIqM I 11-AF ~ u~~ - UQ -h4-o 23RE-04S I 311A 135 A 21t-4 laF-Da lpl-oa I -shy
6 5 3L18 57E+A0 MIpftO 719E+00 147E-03I6 533418 3SlFflI HIPE-fll t~fF- Il - 1352-03 - --- ---- - shy
6 qu3U4 1qfl -v 371f(12 29E-n2 l021)36 55l3I 891-03 ShF-n2 7QUF()3 Inro
mawwn END CYCLE NO tOs 55sq SOF II2 phqE-p3 113jF 2 640E0
s s5sU$ SQ5E-o3 143E-n2 2S7E-53 51 E-0451E0np - - --- -shy ss91a 6ghr-03 ot222A3 1 AA II - - -- ---shy
14 55973 AEL-03 IUIiE-02 21)F -13 30E-0
3 5S988 205E-03 - 29E~nl ItbL03 An7E-ob - -- - - - - --shy
3 S6004 13701-03 5U3E03 bPE-03 lunE-062 5600A 59520(I 1 inIn btIIUe-n 5oE-073 lSL23 311h02 tHEn2 - E a2- n h_____________I7P~l7 - 2 A n F
Cr wt
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
-- - - - - -
- --------
- - - - -- ----- ---
-- - - - -
---
-- --- ---
--- -
- - -- -
----
---
-nS4
3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07
Sh =3 xEn
4 56111 1I36-n1 - 2IE-l 17 E-I(j OE- 5 5lbo IfE-13 - - -I IQF-i -- 51E-03 726k-O6
(I 5hlJa) t6jF-3 IlI3R-l3 06-IAE-fl3 ttBEO6shy3 e49a 70OC-Ot IO6w4no - 9JIF-O1 O75E-O5 5 5 7UZ 66F-(12 67qE-1 815E-03 ha6 E-I15 s S699 67E -1t3 uL71En3 2IsE-(u3 - LAJE -05 57U -s 8e ldS $O3 halI-CII 367E-O
6 2 2 ItI I -- -PEnl --- I3RF= 2 - Q9 -un --saaaa = 2-O--4DA PS+O-- --shy =6 doz Pn - - hlh-03 - l E-n3 - E-ob
6 612U2 9A (13 l- n IEE4E-03 shy 351E-u u-- 1- w ww
5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07
9 61 2T2 2PSE -03 - - sloE-O 11-0 - 2 03 -j913 n a 6l AiU5 l57F-CI I9E-(3 3 175E-03 l n5-ou 11 6I1Ph7 ISAFP-01 3FQE03 67 4E-04 IU0 3 61h83 470Of1 53Et 04 81 E-0l 1E-ob
1- 1 Q T7FL5fp - 5QQEI- l4P-A2 rsok-nshyu 62Qod IOPV-nl IhiE-n3 1I32E-03 1EO ua - A07l - POE-04 IWt -03 -- l(SE-03 - 2P2E-11 3 62l146 216fF-1U 2E0 34I9E-04 537amp--1
-3 -- b2 Ina 7$UE(104 430F riO 18 H e03 364tF -07-shy2 12111 131E0LJ 243E-04- 13O0EQ 93HE-nil 3 $12 - aPE03 L8 7-C3 9AI)F413 22IE-07 S 0P137 -- 17 0 E-03 - 374P-3-- 134E-0O3 - 107F07 P o - q O 6 1 I7Fr4 - I- 3E (I-- l- - -162EA1 h7t~
3 0) Io IJhnF01 33F- n3 P25SE-03 9pqFOA 3 6) 72 aULok-0flU-0 M33j U43nlF-OU 2 i2I7b - s7oEt-l h77E-6S 9blE-05 17E-08shy1 63 IQ I 5t I Ful h11PE 00l fi5qFOpaq -0I h 2A7 lbuP-0 2boE-Ad - lriJF-114 Uffl
70 - Ah 27 0 RE-a3 ~I iSE12 - qh7E-03 shy 84IL3E-07-M 62 332 - QiI 04- 19 1E- 03 - 790 -0~b uQQE-07 a 62Q1 5qF-J iSEnq 3pQ0uQ 330-07
S 6)haU j~nFlP 293L-01 250E-111 PSRF-Ds-shy5 h2945 ui3lII3 I iA-02 87IF-04 4A-0
q 63 145 IQIAF-(4 7204-n4 I1Ak-lh omoEflnik
h 612145 hUAr-oI Q19nEhFOI1 - 3EOa ushy6 65 ls 1 Ii[12 694E-02 qghE-03 2IQFa-041 h 6I1U5 A73FOIS 1b-(03 975E-04U 11010IJ b71U5 649 4F-04s OIE-113 2PhE0U 41E
S 0195 811E04n 3113E114 I9ME-03 331E-0S r 67hL3 Sb~rF0U Iqob-nI - 79- 21S3t-05
LI 077 240R~0U IIE-nU S34JEOU0 U1$F-0A u b 77071 3J3u R(I4AfEAuJ 1~76EmOI4 20l3F06
3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1
La 67910 _plusmniL3Eft4- 1170-EflQu2 It3 IEI04 - -lO E-n7
Ln
-
- - -
-
- - ---- --- -- -- -- oN --- Y--E -O --- shy
- -
e wEND CYCLE NO
-
-
-
----- -- - ------shy
- -
-
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
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0 b
1
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-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
- - -- --- - - -
- - ------------
-- - - - - - - - - - - ----------------
OUTPUT NO 4IO LAO
oSOOTHING rACTaP = 5000 THEII-571 03WEIGHTED IHOOTHINO Aol -28
OPEATO 119ooooXAr
USUAL CAVLCSIamp o
AC 1- NUMbEA J7 10 7~H0t P REfaetwTs ~ o 4 t10c EL rM
3 0 100 0-00 1 0
p-Io0 L
fozzcz o 0-
1 ~-- ------------ 0
I-COZZZ 0000
f 5 N w O - C f ~ I C h ~ b 8 b O b ~ b ~t
~~~ -
-------------------------
V0
---- -C -0 c0
Ph t -- -- - - --shy
------- -~~- -0- - -- -5- - N
lz 11 1
IN1 - --
0 b
1
I3
-- - - - -- - - - Z az tI-Z3zz
]t-- --- ----0-shy0 0 0
-2fshy
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515
- r r-1R7-tlJrr - -r - InFfl --
--- 4~E~ q1 3 iPOn a 0 I a E u 2 - 6 2 Eshy- 0 E ~b - amp S Ft--7- E - O 6 shy
----U~ -- -- ---
C 7U3EO2b-R -67t723 6- - - shy
6d73 Abqna l732-na2E-Ou 339Ej04I7Q9-P4 A40-RM 2-5E-06E-05--
AQ73 15A P 34Etl P F -11 365Enr -
II 5E-fl2 - oqiAE-OM 279E05
C bRA7P3 QO-0
6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5
shy
6 It723 iL17E-114 28UEpu e 7 1A~nU7f 31EflL 112E-04 190E-05
CYCLE NO 1i
a CYCLt Fo ACCtEFF 2Qb77CYCLF NO 7q2 ACCampFF o42CYCLE NO a CLF EFF a77 ACCE FF F 9 CYCLE kO a 5YCLE EFFP
5 CYC E EF 77 ACCEFF 5b6 ---shy- hOCYCE -- CICLE NO 7 CYCLE FFF A 2 CEPFF b2UCYCLE No 6 cYrLF EF 7A ACCEFF 5b
- CYCLE N0-O-CYCLE EFF i CCFF 66A
n - - ----- - - - CYCLE NO 12 YL E~ 7 30 CCF q shy _ 7 2 CCEFF 82
- CYCLE NO 13 CYCLE EFF
--0
tWOn
1515