ASPECTS AND APPLICATIONS OF THEdifferential equations. nathan dinar. college of william and mary....

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


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


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


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)


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


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


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

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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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--- -- 0 13 CYCLE EFF 6s ACC FFP h2 1-- CYCLE

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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

2m 653F~lh I 41114a1 UlItE~fb u7E0 --+ -- -

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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

-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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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

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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

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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

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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

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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

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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

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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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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- CYCLE NO 13 CYCLE EFF

--0

tWOn

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



By Nathan Dinar



SUMMARY




I INTRODUCTION







2



















the finest grid







in coarser grids

3








a coarse grid


















4


analysis)







methods












a system



5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


3 -

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715E-03 773E-03 646E-03 -

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415E-03 307E03

j429E-03 302E-03

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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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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1983-_ 11984 -shy - - - --


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

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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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-

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2 l- - --- - shy

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OUTPUT NO 2 Yd





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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





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9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515



By Nathan Dinar



SUMMARY




I INTRODUCTION







2



















the finest grid







in coarser grids

3








a coarse grid


















4


analysis)







methods












a system



5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

2



















the finest grid







in coarser grids

3








a coarse grid


















4


analysis)







methods












a system



5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

3








a coarse grid


















4


analysis)







methods












a system



5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


L| QL tE-I 2 -

9qt

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2

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END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


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CYCLE NO 1i






--0

tWOn

1515

4


analysis)







methods












a system



5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -

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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

5












The equations are

Sx-Vy fl (xy) (i)

Ux+Vy = f3(xY) (2)

for 05xsl Osyll


U(Oy) = U0 (y) V(xO) = V0 (x) (3)


Figure 1

y V (x)

U (y) iU(y)

v0 )x)

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

6










Figure 2a

VVkV

U U2 f3 J3 U

U U V U

V V V



71 P jJ3t 1-(v2 V1 J



1u 3 -U 2 +V3 vk = f3 (6)



7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

7













on this point






8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

8


u2+

C


2

4f3 U-1i-+








remains unchanged







9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-shy

-

- -


____ _ _I (shyMi3(l

I97(I-OAA -I

- tEi3 77fr~hRV

- -shy lI7PE-l t2

o -7It-I

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Q

1117E-02a

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77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

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1~

71 7 - P - 3E n 7-

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123il

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-


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1033 V-IO-flP


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5 __

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1-

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1I 11~75 431

3 uS- 14

-

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(E O G5E-03

b11 q)F6113

-

-

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t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

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1114E-03 At~IIE-fI4

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__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

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S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

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53-f30E~-tp7 EI h-S 6 - b --S _ n -

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~ ~ -

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9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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-


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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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

|75FfEshyl 2u7F+Aj

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7AI -OU

3

2

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+10

6137E-llO

IP~E-nlshy

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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


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

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2E-nl 234h4-00 i 1 E+10

VE-02 SPSF-03

01

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37328

-5 u0--q 257E-C0-2tqFAl

shy I62EA I


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 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


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

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-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

9



















3method


Output No 1

22 Stokes Equations


AU-px = f (7)

t1 V-py = f2 (8)

U +V = f3 (9)

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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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- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

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123il

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14 -13-

-shy

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~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

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-

__

5 __

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it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

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



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

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2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

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53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

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___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

10







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_













11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

11







Figure 4II












12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

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--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

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--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

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S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

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III1 IF-415 7 4 (F-)6_

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PPF-1 I- r-n5


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53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

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___

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3)-I7E-05 211 E-rIS_ -

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_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

12















Output No 2



[3x(12)

LxR~~x1 1 Vtj~- 21)(13)

)X C (14)




13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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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rog AcLt

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CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

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OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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1 wEo I A E P3 I115E+03 403E-013 6)67

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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

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- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

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----

---

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- -

e wEND CYCLE NO

-

-

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- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


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---- -C -0 c0

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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515

13

















a is defined byx

_i )(16)

I-- 1

21 = LRU(xy)and




14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

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- 17F--lj - 1171F1(p


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1I 11~75 431

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S- F- 5 1E-I5 - RQ1U-s 4AI











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9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


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

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32E-07 1 t -7-shy306E-OA

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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

14



MK~ue1) - P F(tI (8

(19)

(20)

(21

-k~U lRk (22)

4 4shyc-+(23)





(9(1) and N)


figure 5



115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515


115

Figure 5

Vt C




coefficients
















16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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77SF-l _

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It7 + -II -

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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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

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CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

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A

J01I

123il

I-J

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-

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l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

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5 __

-1

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1I 11~75 431

3 uS- 14

-

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(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

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1114E-03 At~IIE-fI4

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__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

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S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

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2ho17 933


- 71E-IA - 4t-E-16

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53-f30E~-tp7 EI h-S 6 - b --S _ n -

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- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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--

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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

-

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

16







technical













No 3






17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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

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_ 100E-12 222E-05

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rog AcLt

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CV(FcentrActOI

-

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2 l- - --- - shy

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OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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-- - - - - -

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-- - - - -

---

-- --- ---

--- -

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----

---

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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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- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


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CYCLE NO 1i






--0

tWOn

1515

17













algorithms













18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




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a 5 9tr 9

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2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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3301 l$4 -Ee7 31 Q 5 7E 7 -

15deg90 2 1

jqcF-f7 7 3E- 117

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

18














simple problems







number of unknowns


iterative cycle



19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



77SF-l _

-shy_4f)

It7 + -II -

-)-11

-shy---

--


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


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

luV o-n IphEI I

tbcoF-112

- r

1 1 -I _ M -

-- - --shy

-

-

-shy

--shy

-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

__- 1

_ 1

A15

- - 411I _ 1 I

CtE

AP

77 E-01 tl I Zll

74I-O1

- 1 iFflt 5q 3E-113

310-o

-

QAF (3p HF b

1~

71 7 - P - 3E n 7-

-

A

J01I

123il

I-J

14 -13-

-shy

-IO -shy

~Q3SI)11

-


-

-

j4-EM -

l-_h 13 31 __

1033 V-IO-flP


II~Q14PEeB2

0Et1

-

__

5 __

-1

it_ --shy

1-

~ shy4 3S

1I 11~75 431

3 uS- 14

-

-shy


(E O G5E-03

b11 q)F6113

-

-

-

t 77E-03 3POE-11

P1-PE-flIS 1 71I1E-05 01 6EO04

-A1

-

-

4AE-i3

5U7Ed-3

3$AEE-a

P65E-113frII

1114E-03 At~IIE-fI4

-

-

- IMq -1a

1 7P)F-1)

Ij

R-1 I bh-f-3

--II~~~~~j _ J$0 7t



















__ 1 E5 - l-501amp-05 q F- C - - 6k-)1 -- E shy


__ P)- 7 P aF-I hb lIE-b7 - 301 F-qh - - -


P 21 012 - j7)E) - 1()5 39E-115 77 E-Pt11

-06 shy

S- F- 5 1E-I5 - RQ1U-s 4AI











Ul



-- -I

-

_ I -bS4 -

2ho17 933


- 71E-IA - 4t-E-16

E-(Ih -

III1 IF-415 7 4 (F-)6_

33E-16

PPF-1 I- r-n5


_3 I

~2h ~b ~lQ~rI$7 ____


53-f30E~-tp7 EI h-S 6 - b --S _ n -

+

I rE-4 ER -~


-

-

- shy


___

___ _ 6 Pi PAORJ

3)-I7E-05 211 E-rIS_ -

-1j-1307V-ol


~ ~ -

-

E -




_ 1J A IPE-17 I r-7 QbbF-119 I E-(7 - -


- 1 2 - 3 1FIS_+( I S5r 9F-0 3IF-lA ---- P - 0 h


9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

1F- sr-n9E

--

-


____ ____

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

-

- -


~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

19










for all amp)






4ncz-~(25)

where

d eE CA V- e0(26)



function A amp

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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A711E IP I $RE-III

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SE-dnl-shy Fl-F I

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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

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~~~_c9 E-O7 2-71E-nt - - I9fl -07 - 471E-117 - -shy

1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

|75FfEshyl 2u7F+Aj

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3

2

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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


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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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

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37328

-5 u0--q 257E-C0-2tqFAl

shy I62EA I


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


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3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

20


factor 7 is found











for example



the computer



stable ones









21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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e wEND CYCLE NO

-

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


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CYCLE NO 1i






--0

tWOn

1515


21



theory



Table 1



1 4 661 6381 5 648 6342 3 680 6952 4 710 125 3 1 714 722







PARABOLIC EQUATIONS



22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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

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I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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1983-_ 11984 -shy - - - --


1249E-03

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_ o509E-04

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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

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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

-422E-05 o687E-05 102E-04 147E-04 988E-05



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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

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3

3 3shy

31105

R10 9a 3 1 L

17J+00

jF4(MO SAIE-nI

13A E+Wl


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

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3 3

37114 7U3)


- 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


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



437V

417 1A 121I Fnj

84hpnl

7 -t2

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71 LF0 I IE-nS

--

3 3

P

a3i7t 1

413 7 60 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

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-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

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AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515

22




























23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


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454(t -shy -

SE-dnl-shy Fl-F I

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-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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Ul



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2 )Il 23 31

3 3 A+tIr-f77E-7


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3301 l$4 -Ee7 31 Q 5 7E 7 -

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

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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

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3

3 3shy

31105

R10 9a 3 1 L

17J+00

jF4(MO SAIE-nI

13A E+Wl


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

31 0141-O -1 317070 5750 -ni jq

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6IE II

pj]lon161)F4041

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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

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37328

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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

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-

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37361 3717 373Q5 37398 -


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37taA 37 555

71sE 0 Isk+0A

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H iMlt+oo IhAE+on


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--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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biiE~fl 177E4nP

0b77k-0l 28 E-0I

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- -

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shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



437V

417 1A 121I Fnj

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423-E-02 -

71 LF0 I IE-nS

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A tIoEI PA 7F-o

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

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e wEND CYCLE NO

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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---- -C -0 c0

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C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515


23





















J tL+Pe X (27)


24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

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a 5 9tr 9

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2

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

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-

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SE-dnl-shy Fl-F I

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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

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Ul



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9~ -it-0O5F rta



1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


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

-

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





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


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


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

- - 6AU2EO IhSFOo

- - PA7E-O 3 7E-0


02f-f5

-shy

- --- ---

3 3

37114 7U3)


- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

-

-

Ul u

37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

24



grids are given by

Ut ~ ~ A ~ ~ shy S1L~~r

L(XA 7) (23)





A s (29)











(



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

- ___



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2

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

-

--shy

-shy

-

-

_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


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AP

77 E-01 tl I Zll

74I-O1

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310-o

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1-

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1I 11~75 431

3 uS- 14

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(E O G5E-03

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P1-PE-flIS 1 71I1E-05 01 6EO04

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Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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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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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


-0

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133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63

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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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-- - - - - -

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- -

e wEND CYCLE NO

-

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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fozzcz o 0-

1 ~-- ------------ 0

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CYCLE NO 1i






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tWOn

1515



t~ (3 1)





represents



For m=l



j 1 k- (32a)







26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


3 -

3-764E-03 107E-02

-622E03

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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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1983-_ 11984 -shy - - - --


1249E-03

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_ o509E-04

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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

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183E-05

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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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rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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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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- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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-- I7E+n3 - 00IF-I




2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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


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


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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


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


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3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02


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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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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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-

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0b77k-0l 28 E-0I

-shy


- -

--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

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----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

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---- -C -0 c0

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C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

26
















approximation







size 64x64)

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

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3

2

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303ia

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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


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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6IE II

pj]lon161)F4041

7021 -01 SM -flil

5 a

3Z$092 39 1)s

274k+AOl 57 E-ntl

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2E-nl 234h4-00 i 1 E+10

VE-02 SPSF-03

01

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37328

-5 u0--q 257E-C0-2tqFAl

shy I62EA I


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

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shy

0F-a06 lQOt-051-0E -2E0

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-

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37361 3717 373Q5 37398 -


-


367F-(2

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- - PA7E-O 3 7E-0


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- 119E-0I9-SEP01i - 6aIaAE1 i97-E-05

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37taA 37 555

71sE 0 Isk+0A

-) 11 7E401 2R15flO

H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

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shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

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-shy

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

27










problems as well














--

28








































Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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

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_ 100E-12 222E-05

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-

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2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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

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133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63

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31 (1660 31 12 AhlF+011

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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


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-

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

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1 ~-- ------------ 0

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CYCLE NO 1i






--0

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Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

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cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


3 -

3-764E-03 107E-02

-622E03

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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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1983-_ 11984 -shy - - - --


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

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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

-422E-05 o687E-05 102E-04 147E-04 988E-05



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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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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



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I 3

3nw 3n677

I48F460 3AIF(10

I h +00 6037E+00

59F+00

65nF+O0 332E-03 173F-tl

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3 3)907 I ft)I

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3

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31105

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31 (1660 31 12 AhlF+011

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4 3

31 0141-O -1 317070 5750 -ni jq

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pj]lon161)F4041

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3Z$092 39 1)s

274k+AOl 57 E-ntl

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1 nno 34 t 3R50 3-657

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50260 93E-02 --shy --shy -shy

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433 O5 357n I3n33

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262P-ll 1

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14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


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e wEND CYCLE NO

-

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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fozzcz o 0-

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--0

tWOn

1515














Tj~(~ Ljuk(33)



L (34)




following

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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1983-_ 11984 -shy - - - --


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

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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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rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

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-

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- - -



2 l- - --- - shy

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OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -

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- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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-- I7E+n3 - 00IF-I




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


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


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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 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

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3 0 100 0-00 1 0

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CYCLE NO 1i






--0

tWOn

1515

30

L2 hU = f+ 2h (35)










the following

L (36)



cVk +()(7



of 33)

2k 3 (38)

ORIGINAL PAGE IS

OF poop QUALITY

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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183E-01 aonE-02

I 3

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I48F460 3AIF(10

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59F+00

65nF+O0 332E-03 173F-tl

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659L400

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3

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31105

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17J+00

jF4(MO SAIE-nI

13A E+Wl


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

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3Z$092 39 1)s

274k+AOl 57 E-ntl

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37328

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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

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0F-a06 lQOt-051-0E -2E0

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37361 3717 373Q5 37398 -


-


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37taA 37 555

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H iMlt+oo IhAE+on


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--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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biiE~fl 177E4nP

0b77k-0l 28 E-0I

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shy

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31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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

-4u7 - 6 E-ot 03070 2op-ol

-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

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IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



437V

417 1A 121I Fnj

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423-E-02 -

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9AhE01 031F02 3276fl

lqE-05- 1 06E

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3 a3 at)1I 13tl

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----

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

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- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

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- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

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- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

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V0

---- -C -0 c0

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------- -~~- -0- - -- -5- - N

lz 11 1

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1

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]t-- --- ----0-shy0 0 0

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C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

31






conclusions





O(h 2 )














ot 06

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




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3 3 A+tIr-f77E-7


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-

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

|75FfEshyl 2u7F+Aj

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659L400

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3

3 3shy

31105

R10 9a 3 1 L

17J+00

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13A E+Wl


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

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3Z$092 39 1)s

274k+AOl 57 E-ntl

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01

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37328

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777

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-

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37361 3717 373Q5 37398 -


-


367F-(2

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37taA 37 555

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1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

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0b77k-0l 28 E-0I

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shy

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31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


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6 6

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21gt~E+02 237E+MO -

50260 93E-02 --shy --shy -shy

h 6

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-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

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262P-ll 1

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3 a3 at)1I 13tl

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I 3

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----

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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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C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

32









this purpose









Table 2 LII i- tL


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)


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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- thE -- IE-O 331)E-13


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S- 5 ___43 4 3Aq 9 P97df

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454(t -shy -

SE-dnl-shy Fl-F I

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-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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9rn 1-)2 IAF-F-)

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1__ __I3 R 27I 1E7Ia I k I - oS_ F-7 _-3I_

2 )Il 23 31

3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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--

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____ ____

3301 l$4 -Ee7 31 Q 5 7E 7 -

15deg90 2 1

jqcF-f7 7 3E- 117

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32E-07 1 t -7-shy306E-OA

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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



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S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- 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

I48F460 3AIF(10

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59F+00

65nF+O0 332E-03 173F-tl

-





3 3)907 I ft)I

35 F11-n jV3F+nl

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659L400

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3

3 3shy

31105

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31 (1660 31 12 AhlF+011

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4 3

31 0141-O -1 317070 5750 -ni jq

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37361 3717 373Q5 37398 -


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0b77k-0l 28 E-0I

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31APO 39070

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50260 93E-02 --shy --shy -shy

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1111603 --- - - -



437V

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_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515


Key to table 2




in all grids




















34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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2

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

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-

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S- 5 ___43 4 3Aq 9 P97df

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454(t -shy -

SE-dnl-shy Fl-F I

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-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

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P1-PE-flIS 1 71I1E-05 01 6EO04

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-- -I

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3 3 A+tIr-f77E-7


471E-118 aA(E717rE-+77cE-07 50O -ON0shy

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3301 l$4 -Ee7 31 Q 5 7E 7 -

15deg90 2 1

jqcF-f7 7 3E- 117

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

|75FfEshyl 2u7F+Aj

3lhF01

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3

2

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+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


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

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274k+AOl 57 E-ntl

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VE-02 SPSF-03

01

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37328

-5 u0--q 257E-C0-2tqFAl

shy I62EA I


SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

777

-shy-

shy

0F-a06 lQOt-051-0E -2E0

-

-

- -


37361 3717 373Q5 37398 -


-


367F-(2

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-

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37taA 37 555

71sE 0 Isk+0A

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H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


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--shy

shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 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

-4u7 - 6 E-ot 03070 2op-ol

-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

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-shy

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

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PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

34






















(39)


for



4

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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77SF-l _

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2

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

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--shy

-

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-shy

-

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_

S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

- -shy ------

-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

6-1 E-02 7PEshy

p I - Q7r -I


9rn 1-)2 IAF-F-)

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Ott G i CL F CT o

o shy - - -

00

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END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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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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


-0

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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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131F-402 23 E n 613AE+02 110TF-n23 1289h -




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133 17 U qbE nn 141 E+4 1 OE4 0E1 64 7E-63

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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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e wEND CYCLE NO

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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CYCLE NO 1i






--0

tWOn

1515

35



Figure 6

Relaxation

LX



















36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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 - -

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


3 -

3-764E-03 107E-02

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I 142E-13 p116E-12 990E-13 8982 -shy - - - - - _ 2 - 774E-04 o113E-03 - 107E-03 8986 -



-

377E-03 267E-03

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1249E-03

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rog AcLt

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CV(FcentrActOI

-

-

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- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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3301 l$4 -Ee7 31 Q 5 7E 7 -

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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-- I7E+n3 - 00IF-I




2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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


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


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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


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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 -




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3 13 13 1 s3 F o| 7u E+nl 7 9E +01 37 E- 02


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

V0

---- -C -0 c0

Ph t -- -- - - --shy

------- -~~- -0- - -- -5- - N

lz 11 1

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0 b

1

I3

-- - - - -- - - - Z az tI-Z3zz

]t-- --- ----0-shy0 0 0

-2fshy



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515

36




d

a -s c

b





Figure 7





point of figure 7








37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

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A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

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454(t -shy -

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-FNn CYCLE MII



P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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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 -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


- h - 7E-Q7 2I7E-n7 1SdE-h7 -- 5Fn








Ott G i CL F CT o

o shy - - -

00

co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




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


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


-0

5 1 2 42 0 M1 F n 2 I b F+ 0)2 I 7 E + 3 18 8F+n t


12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


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


131F-402 23 E n 613AE+02 110TF-n23 1289h -




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


3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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



Z13 36 1 18 E+ n 2 E+ nl S PE - 0 2 1 bO + O 0 5 9 8S













S 24658 2IAF+10 1 171E+ -l 249E+01 F--- -- shy













6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

s 31) 721Si--S ~~ ~ ~



-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

-





3 3)907 I ft)I

35 F11-n jV3F+nl

|75FfEshyl 2u7F+Aj

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3

2

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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

R10 9a 3 1 L

17J+00

jF4(MO SAIE-nI

13A E+Wl


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

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3Z$092 39 1)s

274k+AOl 57 E-ntl

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01

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37328

-5 u0--q 257E-C0-2tqFAl

shy I62EA I


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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37361 3717 373Q5 37398 -


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-

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37taA 37 555

71sE 0 Isk+0A

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H iMlt+oo IhAE+on


- --shy

--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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biiE~fl 177E4nP

0b77k-0l 28 E-0I

-shy


- -

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shy

r 5

31APO 39070

IMSE+o l14I7p

-shy 1s68F+00 3A4E- 1


- 1IAQE-03 SE-03

6 6

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21gt~E+02 237E+MO -

50260 93E-02 --shy --shy -shy

h 6

-4u7 - 6 E-ot 03070 2op-ol

-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



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 - -


A tIoEI PA 7F-o

Pshy1Et(I2 -

9AhE01 031F02 3276fl

lqE-05- 1 06E

-997E -olh-shy - -

-shy

_ ___


3 a3 at)1I 13tl

I5aF2 6b5E-fA

)S-l7PLIE)-O shy

PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

-shy





5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

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



----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


C bRA7P3 QO-0

6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

shy


CYCLE NO 1i






--0

tWOn

1515


37






Table 3



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









= 315

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







- -E -I A 1 F - - A I f 1 - - - - - -n - shy




l --------- S3 A ---- - - - - - - - - - - - - -- ---- - - --- - - - - - - -E- - C Y C L E - -- N ~ E --- -

U-


a 5 9tr 9

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77SF-l _

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P___hOhf001 (IMEt 3 I) 11P -

2

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- thE -- IE-O 331)E-13


A711E IP I $RE-III

- 17F--lj - 1171F1(p


-shy F lP 77P - I

-

-

-

--shy

-

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--shy

-shy

-

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S- 5 ___43 4 3Aq 9 P97df

-



454(t -shy -

SE-dnl-shy Fl-F I

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P 0 9 4 -P-_Ppr-lj__ _ | 149Q _ 1 1 -3



p

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p I - Q7r -I


9rn 1-)2 IAF-F-)

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77 E-01 tl I Zll

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END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

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CYCLE NO 1i






--0

tWOn

1515

38




loss of efficiency









39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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 -

__

160E-02 -187E-02 i OE-02

- -199E-02 285E-02 204E-02

6316 7316 8316


3 -

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8957shy 973---------

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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 -



-

377E-03 267E-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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_ 100E-12 222E-05

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rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

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-

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- - -



2 l- - --- - shy

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OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

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~ ~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~t -~4EQCYCLE NOQ IZ -

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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -





L 67 E E- - I8FE- E-I 7 ----- 76 7 -- -shy5 - - - 1 - 37E- 2E - 07 2 113 L17 -- -- -- 4


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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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6 1 1170 6 17 e (p l7 R +113 497F n 77PF+ ni


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12 13 2 31 IE +A 22U E+to2 IQqE 0P 372 F+0 0


3 1 2 46 2 07 E u 3 4 0 1E + n3 3 5 9E 03 lIb5F - 01

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3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


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-

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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fozzcz o 0-

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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515

39

REFERENCES








APPENDIX

OUTPUT NO 1






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


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


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


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- 108E-12 424E-05 927E-13

1983-_ 11984 -shy - - - --


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

-422E-05 o687E-05 102E-04 147E-04 988E-05



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

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2 l- - --- - shy

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OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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1 0 -E-07 17rE117 - 2A2E-117 IRU3-01 -shy -shy -





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Ott G i CL F CT o

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co

OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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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

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

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6 Q i 1 +O +111| 3+ 6 573 E+VS5 5 sE+O qtTu5

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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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131F-402 23 E n 613AE+02 110TF-n23 1289h -




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3 13 15 0 2 5 3F+ n 2 2b E 4op $45 4E+ n2 P IP E n 2 w (


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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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

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3 0 100 0-00 1 0

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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515

APPENDIX

OUTPUT NO 1






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


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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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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- -199E-02 285E-02 204E-02

6316 7316 8316


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 -



-

377E-03 267E-03

128E-03 E108E03135E-03

170E-03 - - 1187T 11939shy


-358E-05

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1983-_ 11984 -shy - - - --


1249E-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

-422E-05 o687E-05 102E-04 147E-04 988E-05



rog AcLt

14e79 14941 14957 1497314977 14980 1498114962 14986 1500215064 15314 16314 17314 574 56 42 q93

CV(FcentrActOI

-

-

- -shy

- - -



2 l- - --- - shy

__

OUTPUT NO 2 Yd





2LT9P1SANA _ET 1011 211)

EPFL3 5 _ 55 3 Q 0 - - L e VEL -J 6 tu( 61 - - - - - - - - - -



F~J shy-- 1F4 J







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Ott G i CL F CT o

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

u


-- I7E+n3 - 00IF-I




2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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


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3 13)IA1 IQOE+A 2511 +P2 2F (12O 0971EiO

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14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


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50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

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---

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----

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3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

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OUTPUT NO 4IO LAO


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USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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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


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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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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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CYCLE NO 1i






--0

tWOn

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OUTPUT No 3










END CYCLE NO tshy





-- S 511lr03 QIAn2 136F$(ILI I E+1 - -


b -r_ 113s 32En03 - 273E+03 157E+01 7251 +112 168k 4(3 3qQE+(O

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-- I7E+n3 - 00IF-I




2 6927 771E+n2 It P6E+113 E+02 -- 37E+QI F+ -A02 40 aE-03

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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


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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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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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37361 3717 373Q5 37398 -


-


367F-(2

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--



1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

-

biiE~fl 177E4nP

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31APO 39070

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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

-4u7 - 6 E-ot 03070 2op-ol

-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

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3 3

P

a3i7t 1

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3 a3 at)1I 13tl

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PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

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048Eshy5U2EitP

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

- - ---- --- -- -- -- oN --- Y--E -O --- shy

- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

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- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

p-Io0 L

fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

V0

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Ph t -- -- - - --shy

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AQ73 15A P 34Etl P F -11 365Enr -


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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






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6 3)n22 I QRqL400 7bE+on 1135E+00 8MQ7-01

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3 3)907 I ft)I

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31105

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31 (1660 31 12 AhlF+011

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0F-a06 lQOt-051-0E -2E0

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37361 3717 373Q5 37398 -


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1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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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

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- 24AE4np I3E+ol -

21gt~E+02 237E+MO -

50260 93E-02 --shy --shy -shy

h 6

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-a


--shy 72iE-01 293E 01

flOE-n2


5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

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1111603 --- - - -



437V

417 1A 121I Fnj

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

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PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

622Ff-)

048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

------shy

----

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

-

----- -- - ------shy

- -

-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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fozzcz o 0-

1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

V0

---- -C -0 c0

Ph t -- -- - - --shy

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]t-- --- ----0-shy0 0 0

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----U~ -- -- ---

C 7U3EO2b-R -67t723 6- - - shy


AQ73 15A P 34Etl P F -11 365Enr -


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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






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1515

s 31) 721Si--S ~~ ~ ~



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183E-01 aonE-02

I 3

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-





3 3)907 I ft)I

35 F11-n jV3F+nl

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31 (1660 31 12 AhlF+011

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31 0141-O -1 317070 5750 -ni jq

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37328

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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

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0F-a06 lQOt-051-0E -2E0

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37361 3717 373Q5 37398 -


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367F-(2

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1 nno 34 t 3R50 3-657

u2sr+o 3kn0 SQQLI

1 2E-nlI -shy


-

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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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6 6

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50260 93E-02 --shy --shy -shy

h 6

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-a


--shy 72iE-01 293E 01

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5 b

433 O5 357n I3n33

- 21060-shytI E-1I

IflQEtll


-shy-

sp5E-oll 7QE-02

262P-ll 1

h0A-3l nqbs-03

1111603 --- - - -



437V

417 1A 121I Fnj

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71 LF0 I IE-nS

--

3 3

P

a3i7t 1

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A tIoEI PA 7F-o

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9AhE01 031F02 3276fl

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3 a3 at)1I 13tl

I5aF2 6b5E-fA

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PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

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048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

Sh =3 xEn





5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

-

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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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1 ~-- ------------ 0

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~~~ -

-------------------------

V0

---- -C -0 c0

Ph t -- -- - - --shy

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AQ73 15A P 34Etl P F -11 365Enr -


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6 7fl73 41qE-01 - 1iEO04 - 7ppEoO5

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CYCLE NO 1i






--0

tWOn

1515

01

2 2

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37328

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SI007- E-0 t 3AOE-03 -IPIE+0 1 lE-O1

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0F-a06 lQOt-051-0E -2E0

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37361 3717 373Q5 37398 -


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--



1 nno 34 t 3R50 3-657

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1 2E-nlI -shy


-

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biiE~fl 177E4nP

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262P-ll 1

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1111603 --- - - -



437V

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71 LF0 I IE-nS

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3 3

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PIAE-02


-shy


I 3

33 A 2 1387

U0hi-Ip 21qEnp

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048Eshy5U2EitP

- 236E-02 I41(T-A6723b-n6

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

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- -

-

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

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3 0 100 0-00 1 0

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~~~ -

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5-73 +o- IEl---- Q969E-noo - 26E-33 - - - shy











50 1 PS3E -( I t











14 55973 AEL-03 IUIiE-02 21)F -13 30E-0



Cr wt

-- - - - - -

- --------

- - - - -- ----- ---

-- - - - -

---

-- --- ---

--- -

- - -- -

----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

e wEND CYCLE NO

-

-

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----- -- - ------shy

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-

- - -- --- - - -

- - ------------

-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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1 ~-- ------------ 0

I-COZZZ 0000


~~~ -

-------------------------

V0

---- -C -0 c0

Ph t -- -- - - --shy

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1

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-- - - - - -

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-- - - - -

---

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----

---

-nS4

3 sh055 I1QE-(3 -- I50E=O3- - 39T d3 76SE07

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5 6IUJ a 39mP-(3 1T3E-P13 QQUF-03 -31E-07











3 bJ785 I17E-013 7VIEC5 - 198F04 1F-1


Ln

-

- - -

-

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- -

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OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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1 ~-- ------------ 0

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~~~ -

-------------------------

V0

---- -C -0 c0

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CYCLE NO 1i






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-- - - - - - - - - - - ----------------

OUTPUT NO 4IO LAO


OPEATO 119ooooXAr

USUAL CAVLCSIamp o


3 0 100 0-00 1 0

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~~~ -

-------------------------

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Date post:	05-Jul-2020
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ASPECTS AND APPLICATIONS OF THEdifferential equations. nathan dinar. college of william and mary....

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