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HIGH-ORDER SPATIAL DISCRETIZ- TIO ON METHODS FOR THE
SHALLOW TER EQUATIONS
Anita W. Tarn
4 thesis submitted in conformity with the recluirements for the degree of Doctor of Philosophy
Graduate Department of Cornputer Science University of Toronto
Copyright @ 2001 by Anita W. Tarn
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Abstract
High-Order Spatial Discretization Methods for the
S hallow Water Equat ions
Anita W. Tarn
Doctor of Philosophy
Graduate Department of Computer Science
University of Toronto
2001
We present new numerical methods for the shallow water equations on a sphere in
spherical coordinateç. In our implementation, the equations are discretized in time with
the two-level semi-Lagrangian semi-implicit (SLSI) method, and in space on a staggered
grid with the quadratic spline Galerkin (QSG) and the optimal quadratic spline collo-
cation (OQSC) methods. When discretized on a uniform spatial gïid, the solutions are
shown through numerical experiments to be fourth-order in space locally a t the nodes
and midpoints of the spatial grids, and third-order globally.
We also show that, when applied to a simplified version of the shaIIow water equa-
tions, each of oür algorithms yields a neutrally stable solution for the meteorologically
significant Rossby cvaves. Moreover, we demonstrate t hat the Helmholtz equation as-
sociated with the shallow water equations should be derived algebraically rather than
analytically in order for the algorithms to be stable with respect to the Rossby waves.
These results are verified numericâlly using Boyd's equatorial wave equations with initial
conditions to generate a soliton.
We then analyze the performance of our methods on various staggered grids - the A-,
B-, and C-grids. From an eigenvalue analysis of our simplified version of the shallow wa-
ter equations, we conclude t hat, when discretized on the C-grid, our algorit hms faithfully
capture the gruup velocity of inert ia-gravi ty waves. Our analysis sugges ts t hat neit her
the A- nor B-grids will produce such good results. Our theoretical results are supported
by numerical experirnents, in which we discretize Boyd7s equatorial wave equations us-
ing different staggered grids and set the initial conditions for the problem to generate
gravitation modes instead of a soliton. With both the A- and B-grids, some waves are
observed to travel in the cvrong direction, whereas, with the C-grid, gravity waves of al1
wavelengt hs propagate in the correct direction.
Acknowledgement s
There are so many lucky stars around me that, at times, I am at a loss as to where
to begin to count!
1 am extremely grateful to my supervisors, Professors Christina Christara and Ken
Jackson, who: throughout the course of my dissertation, have been supportive a d pa-
tient, and offered me valuable advice and guidance. When 1 joined the numerical anal-
ysis group three years ago as a PhD candidate, 1 had forgotten most of what 1 had
learned, or should have learned, in rny undergraduate numericd analysis courses. So:
even though 1 was supposed to embark on a doctoral project, 1 could hardly remember
how an ODE solver worked, or what it meant for a matrix to have nonzero eigenvalues.
(For the numerically-challenged reader, to a numerical analyst, these two topics are like
swimming to a fish.) Christina and Ken helped me out of my predicament with their
patience and guidance. Despite their busy schedules, they were, and still are, always
available when 1 needed help (which happened quite often). Christina's work on optimal
quadratic spline collocation met hods Lays the foundation for t his project . Ken's strong
mathematical background and insights have added much to the strength and quality of
t his dissertation.
Besides numerical techniques, 1 have also learned from my supervisors the virtue
of being persistent and meticulous. The report 1 wrote for my depth exam, ivhich is
the first technical report required in our doctoral program, was full of inaccuracies and
careless mistakes, and it took us half a year of revision after revision for it to take on an
acceptable form. Though the process kvas tedious and a t times frustrating, I learned a
lot, and (hopefully) my technical writing and research skills have improved since!
1 am thankful to Dr. Steve Thomas, who introduced us to the semi-Lagrangian in-
tegration rnethod, and kept us informed of ongoing research at NCAR. Steve has been
enthusiastic, encouraging, and extremely helpful. He has shared his shallow water code
with us, pointed out to me potential pitfalls in various formulations, and helped me
understand how the results of my project may be useful to the atmospheric modeling
comrnunity.
1 am also thanlcful to Professors Wayne Enright, Robert A h g r e n and Rudi Mathon
for serving on my thesis committee and for their comments on the dissertation, and to
Professor John Boyd of the University of Michigan for serving as the external examiner
on my comnïittee and for his very insightful comments and suggestions.
1 am grateful to my parents, without whom the thought of attending graduat,e school
might never have crossed my mind. My father has always stressed the importance of
education, and sent me to the best schools, even when it meant additional financial
burden for him. As for my mother, 1 fondly remember how she used to t a l e me to
the library and read to me when 1 was little. After 1 started school, she checked my
homework every day to make sure that i t was perfect. Now that my work has become too
mathematical for her to cornprehend, my mother is still willing to listen, and frequently
enquires about my progress.
1 am also thankful to my husband, Harold Layton, whose unflagging s u p p ~ r t and
encouragement have made my months of dissertation writing aspirin-free, caffeine-free.
and relatively st ress-free. Being an academic, he tot ally underst ands the obsession one
sometimes feels when eogaged in a research project, and has never once complained about
the long hoiirs I spent in the office. Through his own work, he has inspired in me a desire
to strive for excellence. When 1 am rewarded by research results after weeks of toiling, he
shares my joy and excitement. When 1 feel discouraged, he tells me that every research
project has its setbacks, and that even the best scientists feel discouraged a t times; and
he lets me know that he believes in me. 1 also wish to thank Harold for introducing
LaSeX to me several years ago when 1 was still an undergraduate.
My two younger siblings, Helen and Eric, are rny spring of joy. Helen has been
supportive and understanding, as always, while Eric, who is twelve years old a t t&e time
of writing, shows his support by asking "So, how was s~hool? '~ every night when 1 corne
home. Even though he was not sure how to pronounce the Greek letter 4, he tried to read
rny thesis, and made the intelligent observation that the shallow water equations must
be hard to solve because there are so many variables! It is my possibly biased prediction
that my brother will grow up to be a great scientist.
Very warm thanks to my colleagues and friends - Jin Lee, Wayne Hayes, Vincent
Gogan - for their support; 1 give them my best wishes. Special thanks to Francois
Pitt, the LaTeX expert in our department, who helped me with many tricky typesetting
t ethniques .
Finally, 1 could not possibly have completed this thesis if it were not for the helpful
staff and excellent facilities at the Department of Computer Science of the University
of Toronto. The financial support provided by the Ontario Graduate Scholarships in
Science and Technology (OGSST), the Open Doctoral Fellowship, and the University of
Toronto Fellowship is gratefully acknowledged.
Contents
1 Introduction 1
2 The Shallow Water Equations and Time Discretkation 5
2.1 The Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . . . . 5
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Notation for Discretization S
. . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Semi-Lagrangian Scheme 9
. . . . . . . . . . . . . . 2.4 Trajectory Calculation in Spherical Coordinates 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Spatial Interpolation 1'7
. . . . . . . . . . . . . . . . . . . . 2.5.1 Linear Lagrange hterpolat ion 19
. . . . . . . . . . . . . . . . . . 2.5.2 Quadratic Lagrange Interpolation 20
. . . . . . . . . . . . . . . . . . . . 2.5.3 Cubic Lagrange Interpolation 21
3 3 . . . . . . . . . . . . . . . . 2.3.1 Cornparison of Interpolation Schemes -- . . . . . . . . . 2.6 The Semi-Lagrangian Semi-hplicit Time Discretization 25
3 Weighted Residual and Semi-Lagrangian Methods 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Background 29
. . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Spatial Discretization 30
. . . . . . . . . . . . . . . . . . . 3.2 Incorporation of Boundary Conditions 32
. . . . . . . . . . . . . . . . . 3.2.1 Longitudinal Boundary Conditions 33
. . . . . . . . . . . . . 3.2.2 Latitudinal Boundary Conditions - U;+' 33
. . . . . . . . . . . . . 3.2.3 Latitudinal Boundary Conditions - &,,,+' 34
vii
3-24 Latitudinal Boundary Conditions . VzC1 . . . . . . . . 34
. . . . . . . . . . . . . . . . . . 3.3 Polar Values of the Upstream Functions 34
. . . . . . . . . . . . . . . . . . . . . 3.4 The Weighted Residual Formulation 3s
. . . . . . . . . . . . . . 3.4.1 Matrices in the Longitudinal Dimension 41
. . . . . . . . . . 3.4.2 Matrices in the Latitudinai Dimension - CI:+' 41
. . . . . . . . . . . 3.4.3 Matrices in the Latitudinal Dimension - &+' 42
. . . . . . . . . . 3.4.4 Matrices in the Latitudinal Dimension - V;+' 43
. . . . . . . . . . . . . . . . . . . . . 3.5 Upstream Funct ion Represent ation 43
. . . . . . . . . . . . . . . . . . . . 3.6 Derivation of the Helmholtz Equation 45
4 The Quadratic Spline Galerkin Method 50
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Background 50
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Quadrature Rule 52
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Existence and Uniqueness 53
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Unstaggered Case 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-32 The C-grid 57
5 The Optimal Quadratic Spline Collocation Methods 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Background 61
. . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Elliptic Problem 63
. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Previous Results 64
. . . . . . . . . . . . 5.1.3 The OQSC Methods for the Elliptic Problem 66
. . . . . . . . . . . . . 5.2 Derivation of OQSC Methods on Staggered Grids 6s
5.2.1 The One-Step Optimal Quadratic Spline Collocation Method . . . 73
5 - 2 2 The Two-Step Optimal Quadratic Spline Collocation Method . . 76
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Existence and Uniqueness 7S
. . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Unstaggered Case i S
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The C-grid 52
6 Numerical Results 85
7 Rossby Wave Stability 91
. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Linear Stability Analysis 91
. . . . . . . . . . . . . . . . . . . . . . . . . 7.11 The Continuous Case 92
. . . . . . . . . . 7.1.2 Algebraic Derivation of the Helmholtz Equation 92
. . . . . . . . . . 7.1.3 Analytic Derivation of the Helmholtz Equation 99
. . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Equatorial Rossby Wave 101
8 Cornparison of Staggering Schemes 105
. . . . . . . . . . . . . . . . . . . . . . . . . . . S . 1 Linear S tability Analysis 10.5
. . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Continuous Case 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The A-grid 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 The B-grid 109
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 The C-grid 118
. . . . . . . . . . . . . . . . . . . . . . . . . 8-2 The Equatorial Kelvin Wave 126
9 Conclusions 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Results 141
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Future Work 144
A The Discrete Operators 147
. . . . . . . . . . . . . . . . . . . A.1 The Quadratic Spline Galerkn Method 147
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 The Matrices 147
. . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Spectral Properties 150
. . . . . . . . . . . . . . . . . A.2 The Quadratic Spline Collocation Method 151
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 The Matrices 151
. . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Spectral Properties 153
. . . . . . . .4.3 The One-Step Optimal Quadratic Spline Collocation Method 154
A.3.1 The Matrices . . . . . . , . . . - - . . - - - - . . . . . . . . . . . 154
A.3-2 Spectral Properties . . . . . . . . . . . . . . - . . - . . . . . . . . 155
Bibliography 156
List of Tables
6.1 The convergence results and computationai costs for QSG and LSG with
different grid sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y 7
6.2 The convergence results and computational cos ts for one-step and two-step
OQSC with different grid sizes. . . . . . . . . . . . . . . . . . . . . . . . SS
6.3 The convergence results and computational costs for standard QSC with
different grid sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SS
List of Figures
2.1 An illustration of the semi-Lagrangian method for one-dimensional advec-
tion. The solid Iine is the actual trajectory and the dashed line is the
approximate trajectory along which the advection function F is integrated. 11
2.2 The actual trajectory (solid), which starts a t i" and ends a t T, is approxi-
mated by the geodesic (dot ted) that passes through Z and F. The rnidpoint
of the geodesic is denoted by P. . . . . . . . . . . . . . . . . . . . . . . . 15
. . . 2.3 A schernatic diagram for the geometry of the trajectory calculation. 16
;O 2.4 A schematic diagram that illustrates how F' is computed from F and r in
e q a t o ( 2 9 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 A schematic diagram that illustrates how the departure position vector i
is computed from the arriva1 position ? a n d the rnidpoint P. According
to the parallelograrn rule, r'+ F = 2 cos O P , which results in equation (2.30). 18
2.6 Norrnalized amplitude for t he wave solution (2.33) with k A x = n / 2 as a
function of a, the relative location of the departure point with respect to
the nearest gridpoint, for linear (solid, thin), quadratic (thick) and cubic
(light, medium) Lagrange interpolations. . . . . . . . . . . . . . . . . . . 23
2.7 Phase error expressed as a fraction of kAx, plotted above as a fucction
of a, for linear (solid, thin), quadratic (thick) and cubic (light, medium)
Lagrange interpolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 A diagram of the gridpoints of Ae and As . . . . . . . . . . . . . . . . . 31
xii
3.2 A diagram of the latitudinal staggered grids and associated basis functions. 3 i
4.1 This diagram show the integration partition for the two-point Gauss-rule
used to approximate the inner product of fi,! and pi- The solid and open
circles mark the griCpoints of the A- and &partitions, respectively. (Note
that they are staggered with respect to each other.) The solid and open
circles together form the gridpoints of the integration partition, whiclr is
also indicated by the dotted lines. The crosses indicate the positions of
the Gauss points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1 Reference solution for the pressure field q5 at t = 16; hours. . . . . . . . . 86
6.2 -4 log-log scale plot of the errors versus N (the number of sub-intervals in
each dimension) for di.fferent methods. . . . . . . . . . . . . . . . . . . . 89
6.3 A log-log scale plot of the errors versus computational costs, measured as
the total number of flops, For different methods. . . . . . . . . . . . . . . 90
7.1 A simulation by the two-step OQSC method of a Rossby soliton traveling
irL the direction of decreasing X values on a 12s x 128 grid for 24 time units
(approximately41 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 A simulation by the two-step OQSC method of a Rossby soliton traveling
iri the direction of decreasing X values on a 33 x :32 grid for 24 time units.
Substantial dispersion, caused by spatial interpolation, can be observed. . 104
8.1 A schematic diagram of the Arakawa A-, B- and C-grids. . . . . . . . . . 106
8.2 Group velocities of continuous gravity solution in the A-direction. Note
. . . . . . . . . . . . . that al1 waves travel in the positive direction. .. 107
8.3 Group veloci ties of continuous gravi ty solution in the 0-direction. Note
that al1 waves travel in the positive direction. . . . . . . . . . . . . . . . 10s
.-*
X l l l
Group velocities of gravity waves in the A-direction for the QSG method
on an A-grid. Note that some waves travel with negative velocities. . . .
Group velocities of gravity waves in the &direction for the QSG method
on an A-grid. Some waves travel with negative velocities. . . . . . . . . .
Group velocities of gravity waves in the A-direction for the QSC rnethod
on an A-grid. Some waves travel with negstive velocities. . . . - . . - . -
Group velocities of gravity waves in the 8-direction for the QSC method
on an A-grid, Some waves travel with negative velocities. . . . . . . . . .
Group velocities of gravity waves in the A-direction for the one-step OQSC
method on an A-grici. Some waves travel with negative velocities. . . . -
Group velocities of gravity waves in the O-direction for the one-step OQSC
method on an A-grid. Some waves travel with negative velocities. . . . .
Group velocities of gravity waves in the A-direction for the two-step OQSC
method on an A-grid. Some waves travel with negative velocities. . . . .
Group veloci ties of gravity waves in the &direction for the two-step OQSC
method on an A-grid. Some waves travel with negative velocities. . . . .
Group velocities of gravity waves in the A-direction for the QSG method
on a B-grid. Some short waves propagate with negative velocities. . . . -
Group velocities of gravity waves in the O-direction for the QSG method
on a B-grid. Some short waves propagate with negative velocities. . . . .
Group velocities of gravity waves in the A-direction for the QSC method
on a B-grid. Some short waves propagate with negative velocities. . . . .
Group velocities of gravity waves in the 6-direction for the QSC methoc!
on a B-grid. Some short waves propagate with negative velocities. . - . .
Group velocities of gravity waves in the A-direction for the ont'step OQSC
method on a B-grid. Some short waves propagate with nega~ive velocities.
xiv
Group velocities of gravity waves in the O-direction for the one-step OQSC
method on a B-grid. Some short waves propagate with negative velocities. 122
Group velocities of gravity waves in the A-direction for the two-step O QS C
method on a B-grid. Some short waves propagate with negative velocities. 123
Group velocities of gravity waves in the 8-direction for the tww-step OQSC
method on a B-grid. Some short waves propagate with negative velocities. 134
Group velocities of gravity waves in the A-direction for the QSG method
. . . . . . . . on a C-grid. The group velocities of all scales are positive. 125
Group velocities of gravity waves in the O-direction for the QSG method
on a C-grid. -4s in the A-direction, group velocities of all scales axe positive.126
Group velocities of gravity waves in the A-direction for the QSC method
. . . . . . . . . . . C U a C-grid. Group velocities of all scales are positive.
Group velocities of gravity waves in the 0-direction for the QSC method a
. . . . . . . . . . . . . C-grid. Group velocities of ail scales are positive.
Group velocities of gravity waves in the A-direction for the one-step OQSC
method a C-grid. Group velocities of al1 scales are positive. . . . . - - . -
Group velocities of gravity waves in the 8-direct ion for the one-step O QSC
method on a C-grid. Group velocities of al1 scales are positive. - - - . - -
Group velocities of gravity waves in the A-direction for the two-step OQSC
method on a C-grid. Group velocities of al1 scales are positive. . - - - -
Group velocities of gravity waves in the 8-direction for the two-step OQSC
method on a C-grid. Group velocities of al1 scales are positive. - - - - - .
Group velocity errors of gravity waves in the A-direction for the QSG
method on a C-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group velocity errors of gravity waves in the 6-direction for the QSG
method on a C-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group velocity errors of gravity waves in the A-direction for the QSC
method on a C-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . -
Group velocity errors of gravity waves in the O-direction for the QSC
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . method on a C-grid.
Group velocity errors of gravity waves in the A-direction for the one-step
. . . . . . . . . . . . . . . . . . . . . . . . . OQSC method on a C-grid.
Group velocity erron of gravity waves in the 6-direction for the one-step
QSC method on a C-grid. . . . . . . . . . . . . . . . . . . . . . . . . . .
Group velocity errors of gravity waves in the Xdirection for the two-step
OQSC method on a C-grid. . . . . . . . . . . . . . . . . . . . . . . . . .
Group velocity errors of gravity waves in the 6-direction for the two-step
QSC method on a C-grid, . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation of gravity waves propagating eastward (in the increasing A-
direction) with the two-step OQSC method on an A-grid. Some gravity
waves travel in the wrong direction. . . . . . . . . . . . . . . . . . . . . .
Simulation of gravity waves propagating eastward (in the increasing X-
direction) with the two-step OQSC method on a B-grid. Again, some grav-
ity waves propagate in the wrong direction (in the decreasing A-direct ion).
Simulation of gravity waves propagating eastward (in the increasing X-
direction) with the two-step OQSC rnethod on a C-grid. The group veioc-
ities of waves of wavelengths have been captured correctly. . . . . . . . .
xvi
Chapter 1
Introduction
Weather prediction is a science with a long h i s t o . Its objective is the description
and prediction of the behaviour of the atmosphere, ocean water and sea ice. Climate
modeling, which predicts statistical meteorological quantities averaged over time and
space through simulations, is also important inasmuch as it helps us understand, for
example, the mechanisms of atmospheric and oceanic circulation, as well as the effects of
technological advancements on the atmosphere. The accuracy of weather prediction and
climate modeling depends on many factors, among which are the accuracy with which
the state of the atmosphere is known at the initial time, the numerical methods applled,
and the resolution used in these methods. Weather prediction computat ions are known
to be very time-consuming. Therefore there is much interest in the scientific cornmunity
in studying accurate and efficient methods for vveather prediction. One way to achieve
high accuracy iii weather prediction computations is to consider high-order discretization
methods.
The spatial discretization schemes that are commonly used in meteorological simula-
tions are finite difference schemes, spectral schemes and finite element schernes. Currently
there is some controversy over which of the three approaches is preferable For the integra-
tion of weather models. For instance, the mode1 at the National Center for Atmospheric
Research (NCAR) incorporates the spectral transform method [32], while the mode1 de-
veloped by the Canadian Meteorological Centre in partnership with the Meteorological
Research Branch (CMC-MRB) adopts a variable-resolution cell-integrated finite element
scheme [IO, 111.
The spectral transform method represents the solution of a problem in sphericd
coordinates in terms of spherical harmonics. Since the spherical harrnonics are the nat ural
representation of the solution of a two-dimensional problem on the sphere. the spectral
approach provides a natural solution to a technical aspect of the pole problem, mhich
is that some variables may not be well defmed at the poles. Also, since the spherical
harmonics are eigenfunctions of the Laplacian on the sphere, the semi-implicit Helmholtz
problem is relatively trivial to solve in spectral space. Another advantage of the spectral
method is that, provided that the solution is sdficiently smooth, the rnethod generates
numerical approximations wit h exponential convergence and t hus wit h accuracy higher
than rnost other methods (e-g., finite-diff-rence methods) for the same spatial resolution.
Alt hough the spectral transforrn met hod seems ideal for the spherical domain, it
also has some disadvantages. Assuming an optimal solver is applied for the solution
of the linear system arising from the Helmholtz problem, the computational cost of
finite-ciifference and finite-elernent rnethods applied to the shallow water equations on
the sphere increases quadratically with the number of gridpoints in one dimension (i-e.,
0 ( N 2 ) , where N is the number of spatial subintervals in one dimension). However, the
cost of performing spectral transforms increases more rapidly. In the case of Fourier
tra.nsfarms in the longitudinal direction, fast Fourier Transforms (FFTs) may be used
and their computational cost increases as 0 ( N 2 log(N)). An efficient method for per-
forming Legendre transforms, analogous to FFTs, has not yet been developed. Thus,
the Legendre transforms in the the latitudinal direction are often performed by summa-
tion and their costs escalate rapidly with increased resolution. Moreover, the spectral
method is formally equivalent to a least squares approximation that minimizes the mean
square error ovrr the global domain. This implies that the size of the error is Likely to
be the same everywhere. This may be a serious disadvantage in more comprehensive
atrnospheric models for a field, such as water vapour, for which the average value varies
greatly over the globe. In the case of water vapour, for example, a small absolute error
may be insignificant in equatorial regiocs, but it may completely alter the character of
the field in polar regions [35]. Thus, there is interest in the atmospheric community in
developing alternative high-order numerical met hods.
In this thesis, we present new nuaerical methods for the shallow water equations on
the sphere in spherical coordinates. The shallow water equations, which describe the
inviscid flow of a thin layer of fhid in two dimensions, have been used for many years
by the atmospheric modeling community as a vehicle for testing promising numerical
met hods for solving weat her prediction and climate modeling problems. In our imple-
mentation, the shallow water equations are discretized in t ime with the semi-lagrangian
semi-implicit (SLSI) scheme, which allows large timesteps while maintaining stability,
and in space on a staggered grid with the quadratic spline Galerkin (QSG) and the
two optimal quadratic spline collocation (OQSC) methods. In order to properly capture
small-scale energy propagation, staggered grids are employed in the spatial discretiza-
tion. We extend current work a n OQSC and QSG methods to systems of partial dif-
ferential equations discretized on staggered grids: and t hen combine t hese methods with
the serni-Lagrangian semi-implicit time integration scheme. We show through numerical
experiments that on a uniform spatial grid the resulting errors of the numerical solutions
are fourth-order l o c d y a t the nacles and rnidpoints of the grid.
The next chapter contains background material for this work. We present a de-
scription of the shallow water equations and their discretization in time with the SLSI
method. The rest of the thesis details our contributions. In Chapter 3, we show how a
generic weightecl residual method can be incorporated into a SLSI scheme, and derive
the associated Helmholtz equation. In Chapters 4 and 5, we focus on three weighted
residud methods - the QSG and the ttvo OQSC methods - m d discuss how these
methods should be modified to solve problems discretized on staggered grids. We also
derive the procedures with which these methods can be applied to the shallom; water
equations in conjunction with the SLSI scheme. In Chapter 6, we present numerical re-
sults illustrât ing the convergence behaviour of both methods and compare their efficienc-
In Chapter 7, using an eigenvalue analysis, we show that, when applied to a simplified
version of the shallow water equations, both methods yield stable and accurate represen-
t at ions of the meteorologically important Rossby waves. Moreover, we demonstrate t hat
the Helmholtz equation associated with the shallow water equations should be derived
algebraically rather than analyticaily in order for the algorithms to be stable with respect
to the Rossby waves. These results are supported by our numerical results for Boyd's
equatorial soliton test problem. The mat hematical analysis and the soLiton test prob-
lem are done in spherical coordinates. This distinguishes our results frorn previous work
based on Cartesian coordinates. In Chapter Y, we analyze the performance of the meth-
ods on various staggered grids when applied to a simplified version of the shallow water
equations, and conclude that, when applied to the Arakawa C-type grid, our methods
faithfully capture the group velocity of inertia-gravity waves. Our analysis suggests t hat
neither the A- nor the B-grids produce such good results. This conclusion is supported
by our numerical results in which we discretize Boyd's equatorial wave equations using
the A-, B- and C-grids and set the initial conditions for the problem to generate gravi-
tational modes instead of a soliton. It is generally agreed in the atmospheric modeling
community that the C-grid is the most accurate for short gravity waves. However, our
literature search indicates that ours is the first detailed analysis of the performance of
the three staggering schemes for the shallow water equations. Finally, in Chapter 9, cve
summarize our results and discuss possible directions for future work.
Chapter 2
The Shallow Water Equations and
Time Discret izat ion
2.1 The Shallow Water Equations
The shallow water equations serve well as a testbed for new numerical methods for
weather predict ion and climate rnodeling as they contain the essential features of more
complete models. For example, both the slowly propagating Rossby modes and the fast-
moving gravitational oscillations are present. Let u and v be the wind velocity compo-
nents in the x- and y-directions, respectively, and 4 be the geopotential, which is related
to the atrnospheric pressure. The shallow water equations in Cartesian coordinates are
where f is the Coriolis parameter, and the subscripts x and y denote spatial derivat ives
in the respective directions. The first two equations are derived from Newton's second
law of motion, and are comrnonly known as the motion equations. The third equation is
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 6
the continuity equation, which enforces the law of conservation of mass. The Lagrangian
derivative is defined by
Because the Earth is spherical, most global atmosp heric models in use today are based
on spherical coordinates. To define the equations on the sphere, Let X be longitude, 13
be latitude, R be the radius of the Earth: fl be its rotational speed, and f = 2Slsin6
be the Coriolis parameter, where R and Q are assumed to be constant. Redefine u =
R cos 6dXldt and v = RdBldt, the curvilinear wind velocity components towards the east
and the north, respectively In spherical coordinates, the shallow water equations take
the form
where the Lagrangian derivative in spherical coordinates is
and the subscripts X and 6 denote the spatial derivatives in the respective directions. The
general developrnent of the shallow water equations can be found in standard texts such
as Haltiner [17] and Holton [19]. Since u and v are multi-valued at the poles, we adopt
the approach of Côté and Staniforth [13] and compute the components of the so-called
wind images instead:
To this end, we multiply the motion equations (2.5) and (2.6) by cos B I R, and rewrite
t h e x in terms of U and V as
CHAPTER 2. THE SHALLOW \VATER EQUATIONS AND TIME CISCRETIZATION '7
dV cos 8 sin 8 -+ fU+- R2
+ s + - ( U Z + V 2 ) = O dt cos2 8
IsoIating the nonlinearity of the continuity equation (2.7) in a logarithmic term, and
multiplying through by cos 6: we have
d dt
+.] = O cose-log#+ - cos 8
Along the longitude, Ii and V are assumed periodic, whereas, at the poles (6 = &n/Z),
homogeneous Dirichlet boundary conditions are imposed: U(X, f ~ / 2 ) = V(X, f r / 2 ) = 0.
The semi-periodic boundary conditions on Q are designed to mimic the behaviour
of its spherical harmonic expansions [13]. Along the longitude, 4 is assumed periodic,
mhile at the poles q5 has zero derivative. To understand the derivation of the boundary
conditions for 4 a t the poles, sve expand it in terms of spherical
The associated Legendre functions Pm,n(p) are generated from the Legendre polynomials
using the relation
Now note that
- - B P,,,(sin 6) d sin 6
m ,n d sin 8 de
dPm,,(sin 9) = C a m , n COS 8 d sin 8
vanishes at the poles. Therefore, we impose the homogeneous Neumann condition dQ/BB =
Some of our mathematical analysis in Chapters 4 to 8 is done on the simplified shallow
water equations, obtained by linearizing (2.5) to (2.7) and assuming constant values for
CHAPTER 2. THE SWALLOW WATER EQUATIONS AND TIME DISCRETIZATION 8
the coefficient f + u tan O / R(= f) associated with the Coriolis terms, for u(= u*) and
v(= v') in the
associated with
cvhere
Lagrangian derivative, and for the coefficients +(= W) and O(= O=)
the gradient terms. The sirnplified equations are
du 4~ - - r u + = O clt R cos O=
In the associated sirnplified ptoblem, we use periodic boundary conditions in both the
longitudinal and Lat i t udinal direct ions.
Equations (2.12) to (2.14) can also be obtained by Linearizing (2.9) to (-'.Il), since,
with cos O' assumed to be constant, U and V are simply constant multiples of u and v.
2.2 Notation for Discretization
Before discussing discretization, we introduce some notation used in this thesis. Let
+(A, 6) be an arbitrary function. Throughout, bold letters (e.g. +) are used for vectors of
values of functions, distinguishing them from the original functions (e-g. $). A superscript
(e.g. $") indicates the time-level at which the function is evaluated. A subscript 1
(e-g. G r ) denotes the biquadratic spline interpolant of a function, whereas a subscript A
(e.g. qA) indicates the biquadratic spline approximation obtained by the collocation or
Galerkin met hod. We illustrate this notation for the geopotential function 4.
q5 = 4(X, 0, t ) is the geopotential function, which, together with the wind velocity func-
tions u and v, satisfies the original shallow water equations (2.5) to (2.7).
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZ-4~10~ 9
@' = dn (A, 0) denotes the solution of the time-discretized equations (see Section 2.6) at
time t , and is a function of X and 0, but not t ,
= 6; (A7 0) is the biquadratic spline approximation to +"(A, 8) by the collocation or
Galerkin method. In our formulation, the shallow water equat ions are discret ized
first in time, then in space. While 4" satisfies the time-discretized and spatially-
continuous equat ions, c& is the solution of the time- and spatially-discretized equa-
t ions.
4; is a vector of values of 4% , (t$a)i,j = 6% (qi, rs, ), evaluated at the rnidpoints (qi , re, )
of the grid associated with the discretization for 4.
4; = +?(A, 0) is the biquadratic spline interpolant of @", used in the mathematical
derivation of the optimal quadratic spline collocation method (see Chapter 5 ) .
2.3 The Semi-Lagrangian Scheme
Discretization schemes based on a serni-Lagrangian treatment of advection have generated
considerable interest in the past decade for the efficient integrat ion of atrnospheric models,
since they offer the promise of larger timesteps, with no loss in accuracy, in cornparison
to the Eulerian-based advection schemes, in which the timestep size is limited by more
severe s tability restrictions.
In an Eulerian-based advection scheme, the observer stays at a fixed geographical
point as the world (or the fluid) evolves around him. This scheme retains the regularity
of the mesh as the observer stays fixed, but requires small timesteps in order to maintain
stability. In a Lagrangian-based scheme, on the other hand, the observer watches the
world evolve while traveling with a fluid particle. This technique is less restricted by
s tability requirements and allows larger timesteps. However, since the fluid particles move
with time, the initially regularly-spaced set of fluid particles usually becomes irregularly-
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 10
spaced as the system evolves.
The semi-lagrangian advect ion scheme attempts to combine the advantages of both
schemes - the regularity of t h e Eulerian scheme and the enhanced stability of the La-
grangian scheme. We briefly describe the semi-Lagrangian method. Consider a simple
one-dimensional advection equation in Cartesian coordinates,
where w(x, t ) is a given function representing the wind speed. Let x ( t ) satisfy the differ-
ential equation
As is well-known, it follows easily from (2.16) and (2.17) that F ( x ( t ) , t ) is constant along
any characteristic x ( t ) satisfying (2.17).
We assume that the values af F are known at al1 spatial mesh points x, at time
t,, where x, denotes the m-th gridpoint (x, m A x ) and t , is the n-th time-level
( tn = nAt). The objective is to compute F at time tnc1 = t , + At at the same spatial
mesh points. This is done by integrating equation (2.17) backward from the arrival point
x, at time t,+i to the departure point x, - ~5x2~' at time t , a d taking F(x,, t,+l) =
F ( z , -6xmf l, t,), where b x l f ' is the displacement of a fluid particle in the time interval
tn to tnti .
In Figure 2.1, the exact trajectory x ( t ) along which the function F is constant is
denoted by the solid c u v e AC, where A and C are the departure and arrival points
respectively. The exact trajectory is approxirnated by the straight dashed line A'BC. In
the semi-Lagrangian method, we approximately integrate the advection equation along
the approximated fluid trajectory. Equation (2.16) implies that F is approximately
constant along A'BC. In other words, if we let 6x$+' be an approximation to 6 x ~ + ' ,
then we have
CHAPTER 2 - THE SHALLOW \VATER EQUATIONS AND TIME DISCRETIZATION 11
This is a two-level semi-lagrangim scheme since two time-Ievels are used to approximate
the Lagrangian derivative-
Figure 2.1: An illustration of the semi-Lagrangian met hod for one-dimensional advect ion.
The solid line is the actual trajectory and the dashed line is the approximate trajectory
along which the advection function F is integrated.
From equation (2.18), if we knom the value of F(x, - bxm', t,), then we have an
approximation to F(x,, tn+i). To cornpute the value of F(x, - bxml, t,), we need (a j
the approximate displacernent 6x;+', and (b) if x, - 6xm n+l is not one of the gridpoints,
we need to interpolate F between gridpoints.
Since l / w evaluated at the midpoint B of A'BC is roughly the slope of the approxi-
mated fluid trajectory, the displacement 6 x ~ + ' can be approximated by 6x2' computed
by the midpoint rule:
Equation (2.19) is solved via fixed-point iteration
CHAPTER 2. THE SHALLOW \VATER EQUATIONS AND TIME DISCRETIZATION 12
using wind speed values from the previous timestep as the initial guess:
&z+nc19[01 = Atw(zm, t,) (221)
When (2.21) is used as the initial guess, two fixed-point iterations (2.20) normally suffice
to attain the suitable accuracy.
If the values of w at time-level t,+$ are not known, then they can be extrapolated
from known values at previous timesteps. For example, linear extrapolation gives
where w n represents the vector of values of w evaluated at gridpoints at time-level tn.
If x, - 6x2' /2 is not one of the gridpoints, then spatial interpolation using gridpoint
values in wn+i is used to approximate w(x, - 6x2'/2, t,+$ between gridpoints.
Once 6x2' is Icnown, an approximation to F at the departure point (x, - C Y X ~ ~ , tn)
can be computed by interpolation. Then we set F(zm, t,+,) = F ( s , - 6xmL, t,).
The semi-Lagrangian advection scheme is summarized below.
1. Use extrapolation, such as equation (2.22), to approximate w(x,, tn,+) at each
gridpoint x,.
2. Apply the fixed-point iteration (2.20) to solve equation (2.19) iteratively for the
displacement ~ x Z + ~ . At the k t h iteration, use spatial interpolation to approximate
n+i,[kl 7 ~ ( x m - Jxm /'-7 in++) -
3- If X, - bxm' is not a gridpoint, then use spatial interpolation to approximate
F ( x , - 6~~~~ tn).
CHAPTER 2- THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 13
2.4 Traject ory Calculation in Spherical Coordinat es
The Eulerian approach of discretizing the shaLlow water equations on a sphere s d e r s
from the pole problem if the gridpoints are distributed eveniy over the sphere. Due to the
convergence of the meridians near the poles, physical distance between adjacent nodes
on the same latitude circle decreases as one moves towards either pole. As a result, the
CFL condition becomes far more restrictive n e z the poles. The semi-Lagrangian semi-
implicit method solves this aspect of the pole problem by removing the CFL condition.
That is, near either pole, a departure point may be many grid intervals away from its
arriva1 point without causing a stability problem.
However, care should still be taken in computing a trajectory neac the poles, where
the X spherical coordinate unit vector varies rapidly in space. In spherical coordinates,
a semi-Lagrangian time discretization scheme approximates the Lagrangian derivative
(3-8) along particle trajectories defined by the velocity vector with components
If, for instance, in computing the departure points, the midpoint rule (2.19) is used to
integrate the veloci ty equations (2.%3), then by expanding X ( t n o ) and X( t , ) as Taylor's
series around X(tnti), we can obtain the error of the midpoint rule for the A-component: 2
The error term may become large near the pole, since it contains the third derivative
of A, which can be obtained by taking the total derivative of the A-equation in (2.23)
twice. The same problem may be encountered in computing the latitudinal component
of the departure points. As a result, the assumption that the velocity components at the
midpoint of the trajectory represent a good approximation to the flow along the entire
trajectory may break clown.
This problem can be solved by transforming to another coordinate system when cal-
culating the trajectory. FVe adopt the approach of Côté and S taniforth [12], and compute
CHAPTER 2- THE SHALLOW WATER EQUATIONS AND SIME DISCRETIZPLTION 14
the trajectory in t hree-dimensional Cartesian geometry with the restriction that the tra-
jectory is confined on the surface of the sphere. The algorithm is outlined below.
Consider a sphere of radius R. Let T = (x, y. r ) be a point on the sphere in a Cartesian
frame of reference fked at the center of the sphere. The Cartesian coordinates x, y and
z are related to the spherical coordinates X and B by
x = RcosXcosB
y = Rsin X cos 8
- = RsinB
4
Let + = (x, y, -7) be the wind velocity in Cartesian coordinates. The components of the
wind velocity in Cartesian coordinates are related to those in spherical coordinates by
sin X x = -RU--RVcosXtanO
cos 0 cos X
y = RU--RVsinXtanO cos e
Note that, unlike dX/dt, which varies rapidly near the poles because of the cos 0 term in
the denominator, none of the velocity components in Cartesian coordinates (x, y, or 2 )
has an "essential" cos 8 term in the denominator after we have cancelled the 'apparent"
cos 0 terms in the denominators with those implicit in the U and V variables. (Recall
that U ucos BIR and V I v cosB/R.)
The trajectory is approximated by a geodesicl (see Figure 2.2) using the midpoint
rule. The general idea is to iteratively compute a midpoint position vector P frorn the
I.0 midpoint velocity r , which in turn depends on P . At the end of the iteration, the
departure position vector F can be cornputed frorn F and rd .
We begin by calculating the Cartesian coordinates x, y and z of each gridpoint T,
given in spherical coordinates, using (2.25). Each midpoint position PJOI = (X0@1, 0'Jo])
'A geodesic is the shortest Iine between trvo points on any mathematically derived surface - a sphere in this case.
CHAPTER 2. THE SHALLOW WATER EQUAT~ONS AND TIME DISCRETIZATION 15
Figure 2.2: The actual trajectory (solid), which starts a t r" and ends a t F, is approximated
by the geodesic (dot ted) tha t passes through 7 and F. The midpoint of the geodesic is
denoted by rd.
is initialized by setting it equal to the associated midpoint position of the previous time-
level. Then the midpoint wind image components un+$ (xO~[OI, 8°-[01) and vn+f ( x ~ ~ [ ~ ] , 0OJ01)
are also initialized. The wind image components un"b and Ifn+$ are first estimated a t
gridpoints by a linear extrapolation of wind images a t time-levels t, and t,-i as in
(2.221, theo approximated a t the initial midpoint position (XOJO], B O * [ ~ I ) through spatial
interpolation. At the k-th iteration, the wind velociiy
a t the rnidpoint of the trajectory is cornputed from (A'@-'], OOJk-'l), un+ f (xo*[~-~] , t9°Jk-11)
and V ~ C * ( X O J ~ - ~ I , B O J ~ - ~ I ) using (2.26). Shen the midpoint position vector is câlculated
( p l = at~r"~~"l LR
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZ-4~10~ 16
Relation (2.28), depicted in Figure 3.3, can be vierved as an application of the midpoint
rule to calculating the arc length from F to r' (Le. ~ R O [ ~ ] ) assuming that the particle 9 ,[k]
remains on the surface of the sphere and has a constant velocity of magnitude Ir 1. A
schematic diagram describing the computation (2.29) is shown in Figure 2.4. Once the
midpoint vector PJkI is computed by (3.19), it is used to compute (X0Jk], ~ ~ ~ [ ~ l ) , rvhich is
then used to update the wind image cornponents un+f ( X O V [ ~ ~ ; and vn+f ( x o ~ [ ~ ] , 9°?[k])
by interpolating gridpoint values of un+$ and v"+&. The above procedure is repeated.
In our implementation, two iterations sufhce to yield the desired level of accuracy.
Figure 2.3: .4 schematic diagram for the geometry of the trajectory calculation.
The departure position vector is given by
d
7:=2cosO?'-r'
which can then be translated back into spherical coordinates. A schematic diagram
describing the computation in (2.30) is found in Figure 2.5.
The algorithm for calculating the trajectory is summarized below.
1. Translate each gridpoint F from spherical to Cartesian coordinates using (2.25).
2. Initialize each midpoint position vector rd*[o] using its value at time-level t,, and
the midpoint wind image cornponents CI"+$ ( x ~ J ~ I , 8 * ~ [ ~ ] ) and V " C ~ (X0J0], O ~ J ~ J ) , using
linear extrapolation at t ime-levels t , and t,-l and spatial interpolation.
CHAPTER 3- THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 17
3. For R = 1,2, - - - , cornpute each PJk] by the following iteration.
(a) Cornpute each midpoint velocity P'[kl fmm P~[k-LI, B ~ J ~ - ~ I )
un+ k (A O y P - L I dOrP-lI) and V"+~(XO*[';-~I aO.[k-'I), using (2.26) and (2.27)-
-0, [k] (b) Cornpute each angular displacement from r using the midpoint rule
(2.2s).
3 ,@l (c) Cornpute each midpoint position from F, @[Y and î using (2.29).
4. Compute each departure position F by (2.30).
5 . Translate each F back to spherical coordinates.
+J . . + F X î
. .? perpendicular to the plane of paper . -
-O
sin O 7 x q,, in the plane of the paper, [ F x i 1
perpendicular to T
9 Figure 2.4: A schematic diagram that illustrates how rd is computed from F and r in
equation (W9) .
Departure points usually fall between gridpoints and spatial interpolation is required to
evaluate function values at departure points. In this section we compare the amount
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME D~SCRETIZATION 19
Figure 2.5: A schematic diagram that illustrates how the departure position vector F is
computed from the arriva1 position Tand the midpoint rd. According to the parallelogam 4
rule, P + T = 2 cos O p , which results in equation (2.30).
of damping and phase errors introduced by the linear, quadratic and cubic Lagrange
interpolation schemes.
Consider again the one-dimensional advection equation (S. 16) and assume a constant
wind speed of w' > O. Let xk x, - w'At be the m-th departure point. Let p denote
the integer offset to the grid interval in which the departure point falls:
Denote the fraction of the grid interval between the departure point and the gridpoint
to its left by a,
where A x = x,-,+l - xm-, is the rneshsize. Note that by definition O 5 a < 1. The
solution F ( x , t ) is assumed to be the waveform
CHAPTER 2- THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 19
where z z a, k denotes the wave numberand v the phase, also h o w n as the frequency.
Note that the true frequency us can be obtained by substituting (2.33) into (2.16):
2.5.1 Linear Lagrange Interpolation
With linear Lagrange interpolation, the value of Fn at the depwture point is approxi-
mated by linear interpolation between x,-, and x,-,+L:
Substituting the solution (2.33) into (2.35) and canceling out the cornmon factor
po e'("m-~)Az+""A') gives
mhich is useful in analyzing the amplitude and phase error introduced by linear inter-
polation. The stability of the numerical solution can be assessed by t aking the absolute
value of each side of (2.36):
Since O 5 a < 1, we have Jl - 2 4 1 - a)( l - cos kAx) 5 1, with equality at cr = 0.
Although by definition a E [O, 1), it is useful to note that equality holds also mhen a = 1.
Therefore the numerical rnethod is stable for any combination of At and Ar, but the
amplitude damps if cr # O or 1 -
Now we study the phase of the numerical solution. Taking the ratio of the imaginary
to the real components of each side of (2 .36) gives
ai sin k A x tan(kpAx + uAt) FZ
1 - a(l - cos kas)
CHAPTER 2. THE SHALLOW \VATER EQUATIONS AND TIME DISCRETIZATION 20
which yields
4 x L a sin kAx v =u -kp- + -arctan ] 3 I At At I - a(1- cos k A x )
where ü represents the phase of the numerical solution.
From the definition of xm and equation (2.32), we obtain the relation
which can be combined mith (2.39) to yield
- Ax 1 a sin kAx u = -kw' - ka- + -arctan
At At 1 - c r ( l - cos kAx) 1 If a = O or û: = 1, then û = -kw' = vx. Therefore, the phase of the numer-
ical solution is correct in this case. However, if a E (O, l), then normally kaAx # arctan [ P sin "z
L -a(l -cos kAx) ] , in which case ü # u', irnplying that the numerical solution suffers
from phase error.
2.5 -2 Quadrat ic Lagrange Interpolation
With quadratic Lagrange interpolation, Fnfl(x,) is approximated as follows:
Substitut ing the solution (2.33) into (2.42) and canceling the common factor
Foe z (k(rn-p) hx+unAt) yields
from which the amplitude of the numericd solution can be obtained. More specifically,
it follows from (2.43) that
1 = le ~(lcpAx+uht) [ z - a2(1 - cos kAx)2 + a4(1 - COS ~ A X ) ~
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 21
Since O < a < 1, J1 - a2(1 - a2)(1 - C O S ( ~ ~ X ) ) * 5 1 with equaLity at CY = O. Although
by definition a E [O, l), it is useful to note that equality holds also when CY = 1. Therefore,
the numerical solution is stable for any combination of At and Ax, but the solution damps
if a # O OC 1. The phase of the numerical solution is
- A 1 CY sin kAx u = -kw' - ka- + -arctan
At At [l - d ( 1 - COS kAx) 1 If a = O or a. = 1, then E = -kw' = v*. Therefore, the phase of the nurner-
ical solution is correct in this case. However, if ol E (0, l ) , then normally k a 4 x # ==titI1 as inkAx
[i-PZ(~-sos k A r ) 1 , in which case G # vX, implying tha t the numerical solution suf-
fers from phase error.
2.5 -3 Cubic Lagrange Interpolation
Cubic Lagrange interpolation approximates F n H ( x , ) using the relations
Substituting the solution (2.33) into (2.46) and canceling the common factor
~~~r(k("-~)Af +"LA') yieldS
where
1 1 I A = ?(a - l ) ( a - 3 ) ( a + 1) - -4% + l ) ( a - 2) cos k A x + - a ( a + l ) ( a - 1) cos2 kAx
3 3 1
l3 = -a sin k 4 x [-(a2 - 4) + (a2 - 1) cos k 4 x ] 3
CHAPTER 2. THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZ- TIO ON 22
The stability of the numerical solution can be assessed by taking the absolute value of
each side of (2-4'7):
By plotting Jm as a function of a and k A x , we observe that J F F F 5 1,
with equality at û: = O and or a = 1. Therefore the numerical solution is stable, but
normally damps. The phase of the numerical solution is
As in the case of linear and quadratic Lagrange interpolation, if a = O o r a = 1, then
G = -kw' = v'. Therefore, the phase of the numerical solution is correct in this case.
Howevet, if a E (O, l), then normally k a A x # arctan [%], in which case ii # vx, implying
that the numerical solution suffers frorn phase error.
2.5.4 Cornparison of Interpolation Schemes
In this section, we compare and analyze the amplitude and phase errors of the numerical
soiutioos obtained by the linear, quadratic and cubic Lagrange interpolation methods.
The amplitude of the numerical solution depends on k A x . Because of aliasing for
values of k 4 s > T , it is standard practice to consider values of k i l x E [O, i r ] only [lJ, 171.
Although the amplitude error is largest at k A x = w; which corresponds to the shortest
resolvable wave, longer waves are numerically more relevant. Therefore, in Figure 2.6, we
plot the normalized amplitudes of the wave solutions with k A x = a/2 for al1 permissible
values of a for the three interpolation methods. Similar results have been obtained for
other values of k A x . Linear Lagrange interpolation has excessive damping, with a max-
imum damping factor of 4 / 2 0.70'71, at a = 1/2. Quadratic Lagrange interpolation
has less damping, with a maximumdamping factor of J5/2 = 023660 at a = 1 / d . Even
better performance is obtained with cubic Lagrange interpolation, which has its largest
damping factor of 5 d / 5 - O.SS39 at a = 1/2.
CHAPTER 2- THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 33
Figure 2.6: Normalized amplitude for the wave solution (2.33) with kilx = ii/2 as a
function of a, the relative location of the departure point with respect to the nearest
gridpoint, for linear (solid, thin), quadratic (thick) and cubic (light, medium) Lagrange
interpolations.
The advantage of the cubic scherne over its linear and quadratic counterparts is more
obvious when we consider the phase error of the three methods. We express the phase
error accumulated in one time interval relative to k A x as
where ü is the numerical phase, v' = -kwX is the true phase, and the [ - ] contains the
quantities in the square brackets in (1.41), (2.45) and (2.49) for the linear, quadratic and
cubic cases, respectively. The phase errors are plotted as functions of a in Figure 2.7
for kAx = ~ / 2 . The absolute values of the phase errors of the linear and cubic methods
have maxima at cr = 0.25 and a = 0.75 and are zero at or = 0.5, while the absolute value
of the phase error of the quadratic method has a maximum at cr = 0.5. The magnitudes
CHAPTER 2- THE SHALLOW WATER EQUATIONS AND TIME DISCRETIZATION 24
of the largest phase errors of the lin=, quadratic and cubic methods are found to be
approximately 0.0452, 0-1257 and 0.0178, respectively. Among the th ree interpolation
methods, the cubic method introduces the smallest phase error. Similar results have also
been obtained for other values of kAx-
Figure 2.7: Phase error expressed as a fraction of kAx, plotted above as a function of
for linear (solid, thin), quadratic (thick) and cubic (light, medium) Lagrange interpola-
tions.
To summarize our results, which agree with the numerical experiments in we found
that cubic interpolation gives s m d l phase error with very little darnping- Moreover, it
is known that cubic interpolation gives fourth-order spatial truncatioc errors. Thus, it
offers a good compromise between accuracy and computational cost, and is the method of
choice in our implernentation. Linear and quadratic interpolations, thougb less expensive,
cause excessive damping and phase errors, whereas higher-order interpolants are more
CHAPTER 2. THE SHALLOW \VATER EQUATIONS AND TIME DISCRETIZATION 25
expensive and the rule of diminishing returns ultimately applies.
2.6 The Semi-Lagrangian Semi-Implicit Time Dis-
cretizat ion
We are now ready to discretize the shallow water equations (2.9) to (2.11) in time using
the SLSI method. In Section 2.3, departure points are defined in terms of trajectories
with gridpoints as downstream points. But in fact, the downstream points can be points
other than gridpoints. In particular, in our formulation, the downstream point (A, 8)
may be a midpoint, a Gauss point of the spatial partition or one of the poles. Therefore,
we generalize the definition of the displacement and the associated departure point as
follows. Let (6XnfL7 6Pf1) be the displacement of a fluid particle in the time interval t,
to tnfL7 ending at the downstream point (A, O ) at time tn+L. The associated departure
point is ( A - 6XnfL, 8 - 6Bnf1) at time t,. For an arbitrary function +(A, 8, t ) , let the
upstream function $"(A, 8) denote +(A - SAnf L, 0 - SPf l , t,), the value of the function ?1>
at the associated departure point a t time t,. The Lagrangian derivative is approximated
b~
Fast moving waves may impose timestep restrictions for stability. In the semi-implicit
time integration formulation, the fast and slow moving waves are treated differently. The
high frequency (i-e., fast moving) modes are discretized irnplicitly, while the low frequency
ones explicitly. To be specific, we treat the gravity and Coriolis terms implicitly by
averaging in time along particle trajectories:
When discret ized in t ime only using the two-level semi-Lagrangian semi-implici t scheme,
the shallow water equetions take the following forrn:
where the correction terrn, bf 2 - + (Pf t)2] sin ën+$ / cos2 ën+k, is evaluated
a t trajectory midpoints and computed explicitly using quadratic extrapolation in time
of the form
on gridpoints, followed by spatial interpolation to obtain trajectory midpoint values. In
equations (2.53) to (2.55), functions a t time-level tncl a re evahated at gridpoints; those
at time-level t,++ are evaluated at approximate trajectory midpoints (Xi - dX;+'/2, O, -
6Oz+'/2); and those a t time-level t , are evaluated a t approximate departure points
(Xi - 6X;+ l , O, - 60;+l). For brevity, let J E pf f and B fif 5 from now on. The
time-discretized shallow water equations c m be rearranged into predictive form for each
timestep by moving the known quantities to the right sides as follows:
~ n + ' Atf uncl + 4 t cos 6 +? U 2R2 6"
- 4t [U4TÇj + hn+l] cos e log Q+' + -
cos 8
To solve the time discretized shaIlow water equations (2.56) to (2.58), the equations
are discretized in space, followeti by the elimination of the wind images U and V from
the system to yield a nonlinear Helmholtz equation for the pressure 4 only. To this end,
CHAPTER 2- THE SHALLOW \VATER EQUATIONS AND TIME DISCRETIZATION 27
we move the $?+' and dsf' terms of the motion equations (2.56) and (2.57) to the right
side and rewrite the motion equations in matrix form
where a , 6, c and cl are functions of 0 and are defined as follows:
Equations (2.58) and (2.60) are then spatially discretized and the divergence terms are
then eliminated from the spatially discretized equations corresponding to (2.58) using
the respective equations corresponding to (2.60) to result in a Helmholtz equation for 4.
The details of the above procedure are described in Chapter 3.
It should be mentioned that higher-order multi-level semi-Lagrangian schemes can be
constructed. Our decision to focus on the two-level scheme is motivated by the general
agreement t hat the time-truncation error is dominated by the space-truncation error.
Consequently, it seerns reasonable to combine a simpIe second-order time integration
scheme with high-order spatial discretization met hods.
Chapter 3
Weighted Residual and
Semi-Lagrangian Met hods
In this chapter, we develop the procedures with which a generic weighted residual method
can be applied to equations t hat are discretized wit h the semi-Lagrangian semi-implicit
method, which proceeds in time by integrating along fluid trajectories. In particular,
we explain how the upstream functions are represented in the biquadratic spline ap-
proximation space, and derive the Helmholtz equation associated with the shallow water
equat ions-
3.1 Background
As will be explained in Chapter 7, the discretized Helmholtz equation is derived from the
spatially discretized form of (2.58) and (2.60), in order to preserve the phase velocity of
Rossby waves. Furthermore, the correct direction for energy propagation is maintained
by discretizing on a staggered grid. In Chapter S we compare three types of stagger-
ing schemes and conclude that the A r a h a C-type grid [l?] has superior performance.
Therefore, in our implementation, the shailow water equations are discretized on a C-
grid. The steps according to which this is done, and how the equations are subsequently
CHAPTER 3- WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN W~ETHODS
solved with the weighted residual method are outlined in this section.
3.1.1 Spatial Discretkation
Along the longitude, we define two uniform partitions that are staggered with respect to
each other by
ivhere AX = 2ir/N\ denotes the meshsize in the A-direction. The gridpoints in ilx and A,\ ..
are chosen so that Xi = i 4 X for i = 0, - - - , !Va\, and Xi = (i + 1/2)AA for i = -1, - - - ; iV-,,
respectively. Similady, staggered partitions As and As are defined in the 8-direction by
.. where 4 8 = a/iVe, B j = - ~ / 2 + jA8 for j = 0, - - , Ne, and B j = - r / 2 + ( j + l /2 )h6 for
j = -1, - Xe. Figure 3.1 shows the gridpoints associated mith As and &. Let {Ï,\& =
(A;-, + ~ ~ ) / 2 ) : 2 ~ and {fii = (ii-, + A;)1.î)2~ be the midpoints and collocation points
N e ive - 1 of A,\ and A,\, respectively, and {T~, = + 0j)/2)j,1 and {PoJ = ( ê j - 1 + 8j)/2}j=1
be the midpoints of he and Ae, respectively. Let {~e,);z~ U{Q, - Bo, TN@+L 3 sNO} and
A - {Fe I )Ne-' 1-1 U{Fg-, - r i / % , q, = iO, îQNe = êp,re*8-l, îeNg+L E ïï/2) be the collocation points
of Ae and h o , respectively. Note that the Ï,\;'s and Ai's almost coincide, as do the î , h
and Ai's, rd, and ê j 7 s , and Po, and Bj7s, but for consistency with literature, these notations
are adopted.
The target functions belong to two-dimensional approximation spaces, chosen to be
tensor products of the associated one-dimensional approximation spaces. Let {Pi(X) =
x - 3 iLX4-1 '$(A ax - i + 2)}:2:' m d { p j ( ~ ) = $$(= - z + T)}i=-l be the sets of basis functions for 2
the one-dimensional quadratic spline approximation spaces corresponding to partitions
Figure 3.1: A diagram of the gridpoints of As and &-
A.\ and A*, respectively, where the model quadratic spline function qû is defined by
1 0 ot herwise B+7:./2 ~ - j + ~ ) ) ~ + L and { j j ( 6 ) = $@(-- For the partitions As and ie7 let {p j (B) = $$J( ,,
1-0 46
j+%)}z'i be the corresponding basis functions, respectively. Figure 3.2 shows a diagram
of the latitudinal staggered grids together with the associated basis functions. There are
N.\ + 2 basis functions associated with A,\, !V,\ + 3 with Ax, No + 2 with A,, and Ne + 3
with as. Note that the latitudinal boundary points (i-e. the poles) are not gridpoints in
4,.
Figure 3.2: A diagram of the latitudinal staggered grids and associated basis functions.
Discretized on a C-grid [17], the target functions, U , V and 4, appearing in equations
CHAPTER 3. WEIGKTED RESIDUAL A N D SEMI-LAGRANGIAN ~ ~ E T H O D S 32
(2.58) and (2.60) are approximated in the biquadratic spline spaces defined on the induced
g i d partitions Al E Ax x As: h2 A,, x A* and A3 G A,, x Ag, respectively, by the
proximated by
Furthermore, we define crin+' - log On+' and approximate it and its upstream function by
3.2 Incorporation of Boundary Conditions
In our analysis and implementation, the basis functions are adjusted so that they satisfy
the boundary conditions by construction. In this tvay, the approximations defined in
(3.1) to (3.4) also satisfy the boundary conditions. In this section, we explain how the
basis functions are adjusted to incorporate the boundary conditions. For a standard one-
dimensional grid partition with N + 1 gridpoints, there are N + 2 degrees of freedom in
representing a function in terms of the quadratic spline basis functions. However, for the
non-standard As, for example, there are & + 3 degrees of freedorn. By incorporating the
boundary conditions, two or three degrees of freedorn can be eliminated. In our case, the
"outerm~st '~ basis functions are removed and the neighbouring ones are modified. Kow
this is done depends on the boundary conditions.
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGR.~NGI.~N METHODS 33
3.2.1 Longitudinal Boundary Conditions
Since the longitude "wraps aroundn the sphere, al1 the functions in (3.1) to (3.4) are
periodic in the longitudinal dimension - a stronger condition than the periodic boundary
conditions. In order to form the periodic basis functions along the longitude, we drop
P0(A), b - l ( ~ ) , ,&(A), ,BNA+l(A) and & + i ( ~ ) 7 and modify the definitions of Pj(X) and
A E [O, X Z ]
X E [ W . ~ ~ - ~ , - ~ I
ot herwise
'$'(A) = , i?j(~)~ j = 9 - Y S . . 7 N,, - 2
,a-i (A) A E [O, JO]
X E [Xi\-3,2ii]
ot herwise
When periodic boundary conditions are applied in the latitudinal dimension, the basis
functions Pr(O) and (6) for j = 1, - - , No are defined sirnilarly.
3.2.2 Latitudinal Boundary Conditions - u'+'
At the poles, Uzfl vanishes: U:+'(X, I ; i / 2 ) = O. TO impose the homogeneous Dirichlet
boundary conditions for Uz+', the modified basis functions P,D(B), i = 1, - - , Ns, are
defined to be
Bf (6) = Pl (Q) - M e )
P p ) Pj(9)7 j = 2 , - - - , I V ~ - ~
0Ee(B) B N , ( ~ ) -Pive+i(O)
CHAPTER 3. WEIGWTED RESIDUAL AND SEMI-LAGRANGIAN METHODS
3.2.3 Latitudinal Boundary Conditions - 42 '
Homogeneous Neumann boundary conditions are imposed on and at the poles,
since their O-derivatives vanish there (see Section 2.1). The associated basis functions
are adjusted to be
3.2.4 Latitudinal Boundary Conditions - vn*' a
The latitudinal homogeneous Dirichlet boundary conditions on v:+' are imposed at the
poles (V-,+'(A, h / 2 ) = O), but the poles are not gridpoints of Âe, since B^i = - ~ / 2 + (i + 1/2)AO for i = -1,. - , Ne due to staggering. Therefore, we cannot incorporate
hornogeneous Dirichlet boundary conditions for V:+' in the sarne way we did for U;+l.
Instead, we first consider the Ne + 3 b a i s functions B j ( 0 ) for j = -1, - - - , !Vo + 1 , as
shown in Figure 3.2. The basis functions are then adjusted to be
3.3 Polar Values of the Upstream Functions
The latitudinal boundary conditions for the upstream functions need to be defined care-
full- Recall that the upstreâm wind image cornponent Oz is defined by
for the displacement (&An+', 66"") associated with the downstream gridpoint (A, O ) for
the time interval t , to t,+l (see Section 2.6).
What do we know about O~(A,fsr/2)? Even though u:+' and Uz vanish at the
poles, the upstream rvind image 0; is not necessarily zero there. To better understand
this, note from the definition (3.5) that
( A , - 2 = Uz(A - SAn", - ~ / 2 - 6BgCL) (3.6)
Ü (A, 2 = Uz(A - GAn+', ii/3 - 68;;') (3.7)
By construction, Uz vanishes at the poles. Hoivever, - r / 2 - &O,"+' and i i / 2 - 60>z1
do not necessarily coincide with the poles; hence, &(A - 6Xn+', - r / 2 - 60n+' 0 and
Uz(A -6An+', 7r/2 - G û ~ ~ ' ) , and consequently Ün (A: &ii/2) are not necessarily zero, mhich
means t hat the latitudinal boundary conditions for Og are not necessarily homogeneous
but general Dirichlet. Extrapolation is used to estimate Ug (A - 6AnfL7 &ii/2 - JO;+'),
where j = O or &, when -ir/2 - 60nCL é [-ii/2, i i / 2 ] or r i /2 - 68;;' 6 [-r/2, si /2] .
Following similar procedures, recall that the upstream wind image component Vz is
defined by
This results in general Dirichlet latitudinal boundary conditions being imposed on Vz. Since #, unlike the wind images, takes on multiple values at the poles, we impose the
general Neumann latitudinal boundary condit ions on & :
- Similar boundary conditions are also imposed on C z .
The modified latitudinal functions introduced in Section 3.2 satisfy homogeneous
boundary conditions by construction. Consequently, the target approximations, O:+',
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN METHODS 36
v;+', #:+' and cf', which indeed satisfy homogeneous boundary conditions, con be
expressed as linear combinations of the appropriate modified b a i s functions:
However, the upstrearn functions, Üx , V;" and <:, satisfi general latitudinal boundary
conditions. Therefore, these functions can be expressed as linear combinations of the
standard basis functions as in (3.1) to (3.4) ; but not of the modified ones in the latitudinal
dimension. Instead, t hey satisfy the following relations:
Since (3.13) to (3.16) are useful in both our implementation and analysis, we d l
derive the expressions for VU, vY, 76" and 7;. We first consider vt . From the definition
of the rnodified latitudinal basis functions for 02 in Section 3 - 2 2 ? @,(O) = ,B?(B) for
j = 9 , ... ,Ne - 1. Rewriting the quadratic spline expansion for Oz (3.1) in terms of
ive - 1 {Dy ( ~ ) ) : 2 ~ and (BY(0) ) j=, yields
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN METHODS 3'7
From the definition of ,OP and ,O$,, we have Pl = Pf +Po and ,ûNe = ,OEe which
can then be used to express the second summation in the right side of (3.17) as
Substituting the above expression into (3.17) and re~rang ing yields
Cornparing (3.18) with (3-13), we obtain
- IV, lV,+l To cornpute qEl the coefficients {U~j}i=;,j=o are first cornputed by applying quadratic
spline interpolation to
using the known values of O: at FA^ , 7-0~ ) ) ~ ~ i ~ ~ ~ L . Then is cornputed using these
coefficients. Similar procedures are used to compute TE, qg and 7;
From the definition of the modified basis functions for v;L+' in Section 3.2.4, we have
and b~~ = ,die + 6BNe+i. Following similar procedures, the quadratic spline expansion
for Vg (3.2) c m be rewritten in terms of { , O f ' ( ~ ) ) z ~ and { ~ ~ ( 0 ) ) ~ o to yield
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN ~ ' ~ E T H O D S
Similarly, frorn the definition of the modified basis functions for 6% in Section 3.2.3,
Following similar procedures, the quadratic spline expansion for & (3.3) can be rewritten
in terms of { / 3 ~ ( ~ ) } ~ ~ and (~j~(~)}z, to yield
,Ne
i= 1
A similar expression can also be derived for ~ 7 :
3.4 The Weighted Residual
In the weighted residual rnethod, the discrete equations are obtained by incorporating
boundary conditions as described in Section 3.2, and imposing conditions on the residuals.
Three sets of two-dimensional test functions are chosen in the following manner. Let
{p'(~))zL and bbe the sets of test functions for the one-dimensional spaces
corresponding to partitions Ax and A,,, respectively. Let {p~(8)}21 and {+jD(6)};io bbe
the corresponding test functions for partitions AB and Âe, respectively, where S = D for
Dirichlet boundary conditions and S = N for Neumann boundary conditions. The sets
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN ~IETHODS 39
The approximate solutions (3.1) to (3.4) ase rewritten in terms of the adjusted basis
functions dehed in Section 3.2, and then substituted into equations (2.5s) and (2.60).
We then set the inner products of the residuals of the resulting equations with each of
the two-dimensional test functions, computed on the partitions associated with the test
functions, to zero. Imposing the above condition on the residuals yields
- ~ i d ( 0 ) 8 ~ ' + (A, 8 ) + ry(h, 0 ) ) $ : (A) (0)dhde
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN ~IETHODS
A
for k = 1, - - - ,NA, 1 = 0: - - - Ne, and
for k = 1, . ,&, 1 = 1, - - , Ne. The residual terms ry; r," and r," arise lrorn non-
homogeneous boundary condit ions imposed on the ups t ream funct ions, and are defined
by
Equations (3.20) to (3.22) can be expressed in matrix form using the rnass and first
derivat ive matrices defined below.
3 .4.1 Matrices in the Lon& udinal Dimension
The mass and first derivative matrices in the longitudinal dimension, where functions are
assumed to be periodic, may be written in terms of the adjusted basis functions as
for i, j = 1, - - - ,1V,\, where wx is a scaling factor independent of X and B. Since both
the basis functions aad test functions are periodic in the longitudinal dimension; and
Pj(X) = pj(X - AX/%), if $:(A) = pT(A - h X / 2 ) , which holds for the test functions for
Galerliin and collocation, then PiVj = Pi,- and p\,,, = Pa\,,, . Therefore, throughout this
thesis, we will drop Pi j and &.,, and use Pij and PAi, instead.
In our mathematical analysis, we use periodic boundary conditions in the latitudinal
dimension as well. When periodic boundary conditions are applied in the latitudinal
dimension, the m a s and fkst derivative matrices in the latitudinal dimension are defined
sirnilarly as the corresponding matrices in the longitudinal dimension, and are written
without a superscript.
In our implementation, as rnentioned before, we use hornogeneous Dirichlet boundary
conditions for U:+' and V;+', a ~ d Neumann ones for dl+'. Next we describe the mass
and first derivative matrices corresponding to the latitudinal dimension and arising in
our irnplementation.
3.4.2 Matrices in the Latitudinal Dimension - U:+'
Io the 8-direction, the integrals may involve functions other than the basis or test func-
tions, such as a(B), b(B) , c(8) or d(8) defined in (2.61). We first define in terms of ,Gy the
CHAPTER 3. WE~GHTED RESIDUAL AND SEMI-LAGRANGIAN ~ ~ E T H O D S
(Ne + 1) x Ne. We t hen consider an arbitrary function g(0) and use the following notation
does not mean function composition.
3.4.3 Matrices in the Latitudinal Dimension - 4 2 '
The mass and f h t derivative matrices for 4:" in the latitudinal dimension are identified
by the superscript N, and expressed in terms of pjv. We first define the matrices
&c- = & SBDB $vw-'V)~~, Qfi, . = & S B , , B ; v r ( ~ ) p N ( ~ ) d ~ ,
~ t y - - J? , @ ~ ( e ) ~ f ( ~ ) d o , Q e b j ~ , h + h - - -L 12 ,o?'(o)+~ (6)dQ we we
for i, j = 1, - - ,Na , and k = O, - - - , Ne. Note that Q" and Q y E l 3 f i X N e whereas
QN.A+& and &FA-'& E nef 1 ) x N e . The associated m a s and first derivative matrices
for in the latitudinal dimension axe defined by
CHAPTER 3, WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN METHODS
3.4.4 Matrices in the Latitudind Dimension - vnf' A
43
ified The mass and first derivative matrices for v:+' in the latitudinal dimension are ident
by the superscript D and the hat symbol, or the superscript -+ 4. We first define the
mat rices
matrices are then defined by
for i, j = O? - . ,Ne, and k = 1: - - . Ne.
3.5 Upst ream Funct ion Represent at ion
The values of a function can be related to the coefficients in its spline representation as
follows. Let Un", rnf', an+' € RN*-N@ and vnf' E R1vx.(N6f '1 be the coefficient vectors
for UE+', CI;+', #kf' and vL", respectively, in the natural ordering beginning with the
n+l A-dimension. That is, u~~=(: - , ) ,~ = Ucj , for exarnple.
Consider the biquadratic spline approximation for the geopotential&+' evaluated at
a midpoint (rAi, 7-0, ):
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN METHODS
Let
. \ i , k = ( \ i ) , E& = p y ( n , )
and E,\ = ((E\.)%=~ and E r = ( E ~ O,,L )lVe jJ=l be the associated matrices. Let $:+' E
R ~ A - ~ ~ be the vector of function values Q ~ + L ( ~ A , , ~ 0 ~ ) in the natural ordering beginning
with the A-dimension. Then
Similar relations hold for the other functions and the associated vectors of coefficients
and values of functions. The matrices associated with LI:+' and viC' are (E,\ @ EF) and
(Es\ @ Ê:), respectively, where
E:,*=PP(~o.), Ê&=Bl%J
and E,D = ( E & ) z = L and ÊB = (ÊD Q I , L )f7!0. J Note that, if linear splines are used, the
coefficients are the same as the function values at gridpoints, and so each of the above
matrices reduces to the identity matrix.
The inverses of (Es\ @ E D , ( E X @ E:) and (E,, @ Ê:) can be iised to compute spline
coefficients from function values. Suppose we want to compute the spline representa-
tion for 4"+' given its values at midpoints. Then it follows from (3.23) that the spline
coefficient vector 8n+' can be computed as
- New suppose we want to expand the function 4: in terms of the splines. The spline
coefficient vector Q" can be cornputed from the values of 4% at the midpoints as follorvs.
First, the values of & a t depârture points (T,~, - S T , ~ ~ , r d , - Sr{+ ') are estimated from
midpoint values using cubic Lagrange interpolation. Cubic Lagrange interpolation is
used instead of quadratic spline interpolation because the latter gives third-order errors
at points that are neither gridpoints nor midpoints. Second, recall the definition of &
CHAPTER 3. w~~~~~~~ RESIDUAL AND SEMI-LAGRANGIAN METHODS 45
where 67;:' 4 a and S r c l denote the particle displacements in the A- and 8-coordinates,
respectively, in the time interval t , to tnC1.l Third, as described in Section 3.3, Sn is
computed by applying quadratic interpolation t o
the following relation:
where I); aises from the general boundary conditions irnposed on &A (see Section 3.3),
and (7)$)i+(j-l)~~ G I);(T,\~, r,, ) for i = 1, . . , and j = 1, + - - , ive. Similarly, Ü z and
can be computed from the midpoint values of Ci: and V,".
3.6 Derivation of the Helmholtz Equation
Equations (3.20) to (3.22) are expressed in matrix form using the mass and first derivative
matrices defined in Section 3.4, and then solved by using (3.20) and (3.21) to eliminate
the Un+' and VnC' dependence in (3.22): and then using the relationship between l? and
(see Section 3.1.1). The result is a discretized Helmholtz equation for &+17 which is
nonlinear due to the presence of a logarithmic terrn. The discret ized Helmholtz equation
is linearized and solved using fixed-point iteration. Once $[;+' is computed, U:+' and
Vif' are updated using the discretized motion equations (3.20) and (3.21). The details
of this procedure are described below. -n+$
Let 6, and &yC' denote the vectors of values of the integrals
' wxwe J J d(0)bn+)(,A, B ) + ~ ( X ) $ ( O ) ~ X ~ ~ and A wxwo J J C ( B ) ~ " + ~ ( X , O)pr(~)+jD(~)d~dO, re-
'Note that with quadratic splines, function values ar midpoints of the spline grid partitions are needed to compute the approsimate solution. Therefore, in this context, we modify the definition of bXf+' and 692' given in Section 2.6 to be the distance traveled by a fluid particle in the time interval in to tn+' whose trajectory ends at the (f, m)-th midpoint (instead of gridpoint) a t time-level -
CHAPTER 3. w-EIGKTED RESIDUAL AND SEMI-LAGRANGIAN METHODS 46
spectively, and T:, T Y and T ; denote the vectors of values of
lf wxwo ~r~(A,O)+~(A)vp(0)dAdO, ~ ~ ~ T ~ ( A , B ) ~ ~ ( X ) + ~ ( B ) ~ X ~ ~ wxwo and
2 W X W ~ J J rE; ( A , O)& (B)dAdO, respectively, arranged in the natural ordering. Equa-
tions (3.20) to (3.22) can be expressed in tensor product form and rearranged as
The next step involves eliminating the UnCL and VnC1 dependence from the discretized
continuity equation (3.29) using (3.27) and (3.25). By pre-multiplying (3.27) and (3.25)
by the inverses of (P @ QD) and (P @ Q ~ ) , respectively; and rearranging, Un+' and
Vn" can be expressed in terms of an+' and other quantities that are known. Then
D,Â+A by pre-rnultiplying the resulting equations by (P,?+~ @ QD O (A)) and ( P 8 Qe )
respectively, we obtain
(P*.+* EI O (A)) cos e un+' = (pfiA 8 Q~ O (A)) COS e (P 8 Q ~ ) - I x
which can then be substituted into (329) to eliminate the divergence term and yield the
equation
(P 8 Q ~ O C ~ S B ' ) P - ~ + ( y ) (P.?+" B QD 0 (A)) ( P B ~ ~ 1 - 1 x cos 9
- ( ) [(P:+" @ Q" O C ) + ( p l 2R2
' '" @I Q: O (dcos B ) ) ] (cn+' + 2") )
= ( @ Q O o n - (F) A [(P?-+A @ QD O (A)) un cos 8
which i s an approximation to a Helmholtz equation, with
(p.?'A 8 Q~ 0 (A)) cos O ( p (8 Q ~ ) - ' (P"~ @ QN O C ) QnCL, and
approximating the second derivative terms (-&)$*:' and (c cos BIQ;;B1, respectively, at
the midpoints of A3. Note that the two terms
D r (p.?'" @ QD 0 (A)) cos 0 (p @3 Q ) 63 Q: a (d cos 8)) an+' and
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN METHODS 4s
almost cancel Throughout this thesis, equation (3.33) is referred to as a discrete
Helmholtz equation even though, technicdy speaking, it is not. Equation (3.32) is non-
Linear because of the nonlinear relationship between ïn*' and *"+' (see Section 3.1.1).
Hence, (3.32) is solved using fixed-point iteration. Following the approach of [34], we
first linearize (3.32) by rnoving the nonlinear term (P @ QN O COS B)rn+l and the known
terms to the right side. Then a fixed-point relation is obtained by adding ( P @ Q")an+'
to both sides. Therefore, at the k-th iteration, w e solve for 9 nf L*[kl in the following
equation, which will be referred to as the linearized Helmholtz equation throughout this
thesis:
{ ( p Q ~ ) - (e) 3R * (P,fi4 @ QD 0 (A)) ( P Bi ~ ~ 1 - l
cos O
- (5) L ( p , t i ~ @ QD O (A)) COS 8 ( p M Q D ) - ~
' ~ n fact, the cancellation wouId be exact if (a) an unstaggered grid were used, (b) the boundary conditions ivere biperiodic, and (c) the functions (*) and d were constants, or, as in the case of collocation methods, could be pulled out of the inner products associated with the Q-matrices. This is because, under t hese conditions, the operators mmmute.
CHAPTER 3. WEIGHTED RESIDUAL AND SEMI-LAGRANGIAN ~'IETHODS
un + P q ~ p + ~ ) ""1 + ,$ - ($) [@+A @ QD 0 (A)) COS 0 (
Note that some variables in equation (3.33) can be computed directly from others.
.4ccording to (3.1), (3.4) and the relation between Cf' and q5*+'? the coefficients vectors
6" and anfL are related to rn and ïn+' by the equations
where (9;)i+(j-1)N* ~F(T , \ , ,T , , ) for i = 1 , s . - ,IV,\ and j = 1, - - - ?.Ri,. At the li-th
iteration, new values for are cornputed, which can then be used to updâte ï n+l.[k]
using (3.35).
When applying the fixed-point iteration (3.33) with an initial guess formed by linear
extrapolation
two fixed-point iterations usuaily suffice. The solution Ont' is used to compute un" and Vn+' using the discretized motion equations (3.27) and (3.2s).
Chapter 4
The Quadratic Spline Galerkin
Method
The quadratic spline Galerkin (QSG) method is discussed in this chapter. FVe present the
steps with rvhich QSG, originally derived for systems discretized on unstaggered grids,
can be modified to be applicable to the shallow water equations: which are discretized on
a C-grid. In particular, we discuss the implications staggering has on the evaluation of
the inner products, and show rnathematically in Section 4.3 that a unique solution exists
when QSG is used to solve a simplified version of the shallow water equations.
4.1 Background
The Galerkin method is arnong the class of weighted residual methods in which the test
functions are chosen to be the basis functions. That is,
where i = 1, - , lV,,, j = 1, - - , NB , and j = 0, - - , Ne. With this choice of test functions,
the residual is orthogonal to the approximation space. T h e scaling factors w~ and we in the
definitions of the mass and derivative matrices are chosen t o be AX and AO, respectively.
That is, the linearized Helmholtz equation (3.33) is solved with the following mass and
derivat ive matrices
in the A-direction for i, j = 1, - - - , VA, and
1 e~ve {QY" g1i.j = lo g ( e ) p ~ f ( e ) j p ( ~ ) d ~ , for i = O, . - ,ive, j = 1;--
1 'ive {Q~v*'" 0 s}ij = lo g ( e ) ~ y ( e ) , ~ ~ ( 9 ) d 9 , for i = 1, . , Ne, j = 0; - ive
1 0% {Q:"'~ O g l i j = lo 9 ( 0 ) , 8 ~ f ( ~ ) ~ ~ v ( 9 ) d 9 7 for i = 1, - ; Ne, j = 0 , , Ne
in the O-direction, where g denotes an arbitrary function of 0 and S = D or N . On a
uniform spatial grid, the error of the resulting numerical solution is fourth-order locally
at nodes and midpoints of the grid and third-order globally, as will be shown numerically
in Chapter 6.
4.2 Quadrature Rule
The entries of the P-matrices a s e simply i ~ e r products of the basis functions or their
derivatives, and are therefore tfme-independent and need only be computed once ana-
lytically at the initialkation stage. The entries of the &-matrices, on the other hand,
involve functions that may be time-dependent. For instance; (QD O a ) , which appears in
r l t i 2 the linearized Helmholtz equation (3.33), varies with time since a = (1 - ( T ) )/(1 + (y)*) , which depends on the depêrture points. Therefore, (QD o a): together with other
Q-matrices that involve time-dependent funct ions, are numerically computed at every
timestep.
In order to preserve the locally fourth-order convergence, the two-point Gauss quadra-
ture rule is used. Care must be taken in defining the grid partition on which the quadra-
ture rule is computed, since t he quadratic spline basis functions, being piecewise poly-
nomials, have discontinuous second derivatives at gridpoints. Therefore, unless the inte-
gration grid partition is chosen properly, the assumption associated wit h the two-point
Gauss rule that the integrand h a s a continuous fourth-derivative within each interval
of the integration grid partition may be violated, and, consequently, fourth-order con-
vergence cannot be guaranteed. In order to satisfy the smoothness requirement of the
two-point Gauss rule, the integration grid is defined to be twice as refined as the spline
grid in each space dimension. T h a t is, the integration grids are hi (A,! U 8.1) n[O, 'a] and A i = (Ag U &) n[-7i/2,ii/2] in the A- and O-directions, respectively. Consequently,
the points of discontinuity of t h e spline basis and test functions do not fa11 within the
integration grid intervals (see Figure 4.1). This complication tvould not arise if an un-
staggered grid tvere used because in that case al1 points of discontinuity of the basis and
test functions would coincide w i t h the spline grid, and consequently, the spline grid couid
be used as the integration grid f o r computing the integral.
When the QSG method is applied to the linearized Helmholtz equation (3.33), the
inner products in the 8-dimension are first cornputed using the two-point Gauss rule as
Figure 4.1: This diagram show the integration partition for the two-point Gauss-rule
used to approximate the inner product of & and 0;. The solid and open circies mark the
gridpoints of the A- and respectively. (Note that they are staggered with
respect to each other.) The solid and open circles together form the gridpoints of the
integration partition, which is also indicated by the dotted Lines. The crosses indicate
the positions of the Gauss points.
described above before the iteration (3.33) is begun. Then a t each iteration of (3.33),
the Linear systern is solved with the conjugate gradient iterative method to yield @ n+i ,[Y
The convergence behaviour of the solution, together with the accuracy and efficiency of
the method, is discussed in Chapter 6.
4.3 Existence and Uniqueness
In this section, we prove that the solution of a simplified version of the linearized
Helmholtz equatioo (3.33), obtained with the QSG method, exists and is unique. To
be more specific, the analysis is done on the linearized shallow water equations with
constant coefficients, equations (2.12) to (2.14), with the additional assumption that the
boundary conditions are biperiodic. Since these equations are linear and do not contain
the logarit hmic term, the associated Helmholtz equation (3.32) and its linearized form
(3.33) are essentiaily the same. Therefore, by showing the existence of a solution for
(3.33), we also show that the Helmholtz equation (3.32) associated with the simplified
shallow water equations is solvable.
The linearized Helmholtz equation (3.33) of the simplified system, derived by first
discretizing the equations and then elirninating the divergence terms from the discretized
cont inuity equat ion corresponding to equation (2.14), t akes the form
where RH is known. We will prove that a unique solution exists for the above equation
mhen the discretization is done on an unstaggered grid, and also on the C-grid.
4.3.1 The Unstaggered Case
On an unstaggered grid, the linear operator Ii is defined as follows:
Note that Ii' differs from the operator associated with the left side of (3.33) in that the
latter operates on a staggered grid and allows variable coefficients, and that biperiodic
boundary conditions are assumed in Ti'.
A unique solution exists for equation (4.1) if and only if Ii is nonsingular. We will
prove that Ii' is nonsingular by showing that al1 eigenvalues of K are nonzero. Since
constant values are assumed for the coefficients in the linearized shailow water equations.
the integrals associated with the mass and derivative matrices involve only the products
of the basis functions or their derivatives, and can be computed analytically. These
matrices are defined in Appendix A. An important property of these matrices is that
P and Px have the same eigenvectors. Similarly, Q and Qe have the same eigenvectors.
Let pk and pal, be the eigenvalues associated with the k-th eigenvector of P and f i ,
respectively, and ql and qo, be those corresponding to the 1-th eigenvector of Q and Qs,
respectively. In Appendix A we show that
- 2 PAk - '- 3 4 X
cos (7) [3 - sinZ
Let
Wit h biperiodic boundary conditions, the second and third terms inside the square brack-
ets in (4.2) cancel out, and the eigenvalues of K, denoted by gk,i, can be expressed in
terrns of the eigenvalues of P, Pd{, Q and Qe as follows:
+ (-)'WC{ I R 9AX2 4 cos 0- ak(l - ak)(3 - ~ ~ ) ~ ( 1 5 - 15aj + h i 2 )
(15 - l5a i + 2a:)
Note that for al1 values of II- and Z
and (15 - 15ak + -a:), 2 2 , (15 - 15~2; + -ai2) 2 2
Since QS 2 O, c 2 O and cos@' > O for 0- E ( - . / 2 , i r / 2 ) , we conclude that o k , r 3
( $ ) 2 > O. This irnplies that K-' exists. Consequently, a unique solution exists for the
Helmholtz equation (4.1).
As mentioned before, Ii is not, strictly speaking, a Helmholtz operator when the
"Helmholtz" equation is derived algebraically, i.e. the elimination of the divergence terms
is dore after spatial discretization. A true Helmholtz equation may be derived analyt-
ically by eliminating Un+' and Vnf' from the space continuous equation (2.58)- If we
follow this approach, the Helmholtz operator TCH becomes
where P,\,i and Qso are the second derivative operators in the X and O-directions, respec-
tit-ely, and are defined in Appendix -4. These matrices have the same eigenvectors as P
and Q, respectively. The associated eigenvalues are
H 1 = pkq~ - 2R [' cos Ox pxxkq~ + cos @lpkqeor 1
4 cos O' 1 + 3 N 2 4 ( 3 - 2 a i ) [z(15 - 15ak + 2 4 ) ] } ( 4 - 5 )
Compâring (4 .3) and (4 .5 ) , we note that $(1- crk)(3 - a k ) 2 / ( 1 5 - 15ak +2rut) 5 &(3 - 4 2 a k ) and $(i - 4 ) ( 3 - ~ 4 ) ~ / ( 1 5 - 15ai +- %ai2) < ,(3 - 2 4 ) . Thus, we conclude
that O < 5 O&, implying that (4.4) is uniquely solvable too. However, as wiLl be
explained in Chapter 7, t his approach of analyticaliy deriving the Helmholtz equation is
not recommended as it tends to darnp the Rossby waves.
4.3.2 The C-grid
%%en the simplified shdlow water equations (2.12) to (2.14) are discretized on a C-grid,
the linear operator K becomes:
The staggering operators PA-)' and P,:" share the same eigenvectors as P and Px,
as do P ' + ~ and ~ , f + ~ . The eigenvalues associated with the k-th eigenvectors of P"+'
and pi+" are e z ' Y p ~ and e-'*pz, respectively, where
I + - P k =
kAX kAX kAX cos (.) [120 - 60sin2 + sin' (T)] 120 -
Sirnilarly, the eigenvalues associated wit h the k-th eigenvectors of and are a x
et'? py k and e-'"pz, respectively, where
In the latitudinal dimension, Q * + ~ and Q$+A share the same eigenvectors as Q and Qs.
as do Q'+A and Q$+A. The eigenvalues associated with the 1-th eigenvectors of Q&+'
and Q'+* are eziYql+ and e-"Yqr>, respectively, where
Similarly, the eigenvalues associated with the 1-eigenvectors of and Q ~ + A are
A6 e t l rq , t and e-t~y qol + , respectively, where
The derivation of the above expressions can be found in Appendix A. The eigenvalues
of f< are
+2
Ck.1 CP' c -- = .r;41-(g)' { [ cos@= pr;
where
d k AX 3 - LA0 +-cos (-) (120 - 60ak + ai) - sin (T)
120 3A0 2 32 - 15(24 - 20at + a:)
AX(15 - l5ak + 2aE) 32c
~ ~ ( 2 4 - - O a k + cr:)(15 - 15a; + -aiZ) [15 cos @"AA
2d sin(k4A) sin(lA0) (120 + - Goa* + cr:)(3 - a;)] 4 - 120 3A0
2 (2; a, = - ( ) a-- ( C C O S W ~ ~ ~ ; - d p A k g ) 41 - sin (q) (24 - 2 0 4 + ai2)
L(15 - 15
150; + 2ai2)
c cos O' 32 (!,O) [ 15 (15 - 15ak + h:)- sin - (24 - 2 0 4 + ai2)
A0 2d kOX kAA
3AX 120 l (?)(120-60a;+a;~) 1 - -sin (-) cos (2-) (3 - a + - - c o r
We study the behaviour of o k , ~ for typical parameter values. Since it has been shown
in the previous section that [&(l5 - 15ak + k:)] [&(l5 - Z5ai + 2ai2)] 3 &+ in order
for 0k.r > O, it is sufficient to show that Al > -& and A2 > -&. For simplicity.
assume that At = AX = 4 0 r h. We first consider AL- Since c > O and cos 0" > 0, the -
first term inside the square brackets in (4.10) is always nonnegative. Since d = y c , the
second term is much smaller than the first for suEciently large an. and sufficiently small
h. When ak = 0, the first term approaches zero. Nonetheless, at crk = 0,
which means that the second term also vanishes. Hence, Ai = O. Also, for small positive
k's, we have a k = sin2(k-h/%) ( k / ~ / % ) ~ . Substituting this approximation into the
expression for Al, and writing d as gc yields
3 2 { 15 cos O=h ( ~ ) 2 [ ? 4 - 2 0 ( ~ ) 2 + ( ~ ) " ] ( 1 5 - 1 5 a { + 1 a ; 2 )
Since the first term inside the curly brackets is positive, rve have
by noting that sin(lA6) 2 -1 and 3 - sin2(?) 5 3. Since f « 1; h « I, R » 9' and
c 1, we have At = (3(h2) . Hence, when crk (kh /3)2 for a srnall positive integer k,
for h suficiently small. -4 > -= Sirnilar reasoning may be applied to A2 to show that A2 > -215 for small l 3 0 and
sufficiently small h. Hence, c k , r > O, in the cases that we have considered. Unfortunately,
this does not cover a11 possible cases, so we have not actually proved that ok ,r > O for al1 k
and 1- However, our numerical results in Section 6 indicate that the matrix K associated
ivith (4.1) is nonsingular mhen the simplified problem (2.13) to (2.14) is discret ized on
the C-grid.
Chapter 5
The Optimal Quadratic Spline
Collocat ion Met hods
In this chapter, we present details on the optimal quadratic spline collocation (OQSC)
methods. The quadratic spline collocation (QSC) rnethod, when applied in its standard
form, leads to suboptimal solution approximations, in the sense that the order of conver-
gence of the spline collocation approximation is lower than that of the spline interpolant
in the same approximation space. To obtain optimal solution approximations, the OQSC
methods can be used [6, %O]. In Section 5.1, we summarize previous ivork done in OQSC
and outline how the methods yield optimal solutions for an elliptic partial different equa-
tion. In the rest of this chapter, we will explain how the two OQSC methods, originally
derived for a single elliptic partial differential equation and later generalized to systems on
unstaggered grids [25], can be modified to be applicable to the shallow water equations,
which are discretized on the s taggered C-grid.
5.1 Background
A collocation method can be interpreted as a special case of the weighted residual method
where the test functions are chosen to be delta functions associated with the collocation
points, which, for quadratic splines, are usually chosen to be the miclpoints of the par-
tition. The scaling factors in the definitions of the mass and derivative matrices are set
to be one: W A = ug = 1. With collocation, the mass matrix P maps coefficients to func-
tion values at collocation points in the same partition, whereas P A-'* rnaps coefficients
defbed with respect to partition A,\ to the corresponding function values at collocation
points of A,]: and sirnilarly for PA+". The first derivative matrix Pi maps coefficients to
first derivative values at collocation points within the sârne partition, while P,\+' maps
coefficients defined with respect to Ax to first derivative values at collocation points in
AA, and sirnilarly for P,\+J- More specifically, we have
for i, j = 1, - - - , Nd\. Note that P i j = Edli,-, where E,, was defined in Section 3.5.
Similarly, the mass and first derivative matrices in the O-direction are defined by
where S = D or N . For the Galerkin method, evaluating the finite element matrices may
require evaluating integrals involving functions other than the basis functions. For the
collocation method, however, it is a lot simpler to compute the Q-matrices since the test
functions are chosen to be the delta functions. For an arbitrary function g(B), we have
the following
{QS O g } i j = g(~~i )Q$y
where S = D or N.
The standard QSC method gives only second-order convergence, which is suboptimal.
To obtain optimal solution approximations, the one-step or two-step correction method
can be used [6, 201. In the one-step method, the differential operators are perturbed to
eliminate the low-order residual terms, and optimal solutions are cornputed using the
perturbed operators. Ln the two-step rnethod, a second-order approximation is generated
first, using the standard formulation, and then a higher order solution is cornputed in a
second step by perturbing the right sides of the equations appropriately.
5.1.1 The Elliptic Problem
The OQSC methods were motivated by the second-order linear elliptic partial differential
equat ion
Cu - au,,^ + buxe + + dux + eue + fu = g
on Cl (ai, bi) x ( a z , b 2 ) , subject to mixed boundary conditions
B u ~ c r u + , 8 u , = y
on 8fl boundary of R, where u, a , b, c, d, e, f, g, a, 0, and 7 are functions of X and 8
and un denotes the normal derivatives of u. PVe will briefly describe the formulation of
the OQSC methods for the above problem. Further details can be found in [6] .
Consider the rectangle fi E R U d o = [al , bl] x [a2, b2], and let Ax and As be uniforrn
partitions of the intervals [a l , bL] and [aZ, bz], respectively. That is, A,\ { A o = a 1, X i =
ai + i(61 - al)/^.^)^^ and As {Bo = a2, ûj = a2 + j(b2 - a 2 ) / & ) 2 , , as are d e h e d in
section 3.1. Let {rd\, = (Xi-l + X ~ ) / ~ ) Z ~ and {Te, = (8 j - l + 0 ~ ) / 2 } ~ ~ be the rnidpoints
of A, and Ae, respectively. For an arbitrary function $(Xoû), let l L i j +(T-\,; r d , ) , for
i = O,.-. ,LVa\ + 1 and j = 0 , - - ,Ne + 1, where r\, = X o - - T4\NA+i = AN, = 61,
re0 = 90 = a2 and T , ~ ~ + ~ = ON, = bz. Let U r , defined on the induced grid partition
ilx x A s , be the biquadratic spline interpolant of a function u E C6 such that
for i = 1, - - , Nd\ and j = 1 , - - . , &,
for i = O or N,\ + l and j = 1 , - - . , N e , and
for i = 1, - , Na\ and j = O or ive + 1. At each of the four corners of S1 tvhere i = O or
iV.\ + 1 and j = O or 1Vg + 1, U I satisfies one of the interpolation relations (5.4) or ( 5 . 5 ) .
5.1.2 Previous Results
The following results in Christara [6] are useful in deriving the OQSC methods for the
elliptic problem (5.1) and (5.2) and also for the modified methods on staggered grids.
Theorem 1 Let be the biquadratic spline interpolant of n funct ion u E C6 as de-
Jned by (5.3): (5.4) and (5.5). T h e n the following relations hold at the midpoints
where DI denotes the k - t h deriuatioe operator wi th respect to the variaHe s, D,ie denotes
the cross-derivative operator with respect to X and 6, and h max(AX, As).
iVA IV, At the points {A;, TeJ}zdiy. l a n d {T,\,, Oj}i=;,y=o of a u n i f o n n partition A the following
relations hold:
Theorem 2 Define the difierence operator A* 6y
for i = 2, . , AJ,\ - 1 and j = 1, - - - , No, and As by
for i = 1,- - - ,iV,\ and j = 2, - - - , ive - 1. I f ur is the biquadratic spline interpolant of
u E C6 deJned by (5.3), (5.4) and (5.5), t h e n
for i = 2: . - - ,fi - 1 and j = 1,-•. ,&, and
f o r i = l , - . . ,& a d j = 2, - - - , i'Vg - 1. At the collocation points close to the boundaries,
the hîgher deriuatiues in the O-direction are npproximated by
for k = 3 or 4 and i = 1,. . , Nd\. Similarly, approximations for the deriuatiues w i t h
respect to X are obfained.
5.1.3 The OQSC Methods for the Elliptic Problem
From the relations in Theorem 1, the differential equation (5.1) and the boundary con-
ditions (5.2), we observe that the interpolant ut of the true solution u of (5.1) and (5.2 ),
defined by (5.3), (5.4) and (51.5)~ satisfies the relations
NA, l'Je a t the points { r i i , Te,)i=l,j=l, and
a t the points (T.\~, T@,) for i = O or Nd\ + 1 and j = 1, - , N e , and
a t the points (ni, TQ,) for i = 1, - , N,, and j = O or Ne + 1 . At the four corners (rAi, Te , ]
where i = O or NA + 1 and j = O or Ne + 1 , uri , satisfies either (5.22) or (5 .23) .
Fourt h-order approximations to the residual terms can be obt ained by approximatirïg
the higher derivatives of u on the right sides of the above equations using ur and the
difference operators. Using results in Theorern 2, equation (5.21) c m be rewritten as
f o r i = 2 , - - - ,Ni\-1 and j = 2 , - - - , No - 1, where 6L is an 0 ( h 2 ) perturbation operator.
At points close to the boundaries, the higher derivatives are obtained by applying (5.1s)
and (5.19). In this way, the dehition of 6,C is extended to all interior collocation points.
More details can be found in Section 5.2, where we present simiIar procedures for the
OQSC methods on staggered grids.
The boundary operator residual equations (5.22) and (5.23) at boundary collocat ion
ive + 1 points { ( X ~ , T , \ , ) , ( X ~ ~ , T , ~ ~ ) ) ~ = ~ take theform
where (k, 1, rn) = (0,2,3) or (Nd\ + 1, N,\ - 1, Na\ - 2), and 6B is an O(hZ) perturbation
operator. At the points {(qi, Bo) , (T,\,, B , v e ) ) ~ l , a sirnilar relation can be derived.
The one-step OQSC method can be derived by moving the perturbation terms in
(5.24) and (5.25) to the left and determining a biquadratic spline that satisfies
at the midpoints of AA x A,, and
at the boundary collocation points including corners. It has been shown in [6] that U A
is fourth-order locally at gridpoints and midpoints, and third-order globally. Supercon-
vergence is also exhibited by its derivatives at special points [fi].
Ln the two-step OQSC method, a second-order biquadratic spline approximation U A
is generated first by forcing it to satisfy
at the midpoints of A,, x Ao, and
at the boundary collocation points. The second-order approximation Ga is then used
to perturb the right sides of the differential equation and boundary conditions, thereby
elirninating the second-order residual terms. By determining a biquadrat ic spline U A t hat
sat isfies
at the midpoints of A,, x As, and
at the boundary collocation points and corners, Ive obtain an approxirnate solution to
(5-1) and (5.2) that is, as in the case of the one-step method, locally fourth-order at
gridpoints and midpoints of the uniform partition, and globally third-order.
5.2 Derivation of OQSC Methods on Staggered Grids
We will extend the methods described above to staggered grids and demonstrate through
numerical experiments that the locaUy fourth-order accuracy is preserved when the OQSC
methods are applied to the coupled shallow water equations, which are discretized spa-
tially on the staggered C-grid. The collocation points for Al = AA x As, A, = hA x Ls
NA .ive NA 1'43 IV, ,Ne and A3 = AA x 40 are chosen to be {i\,, ~ , ) ~ = ~ , j = ~ , {~,\i, % ~ , ) ~ = ; , ~ = ~ 7 and IT\,, rt, )i=l,j=i?
Li
respectively. Note that the collocation points +O,, = Bo and ieNa = B N ~ - ~ are gridpoints of
As (see Section 3.1.1).
Let u;" and 0; be the biquadratic spline interpolants of Un+' and Ün, which are the
true solution and its upstream function for the time-discretized shallow water equations
(2.58) and (3.60), respectively, defuied on the partition AL (see Section 3.2.1), such that
the relations
are satisfied for i = 1, - - , !yx and j = 1, - , lVel and
for i = 1, - - - , N , and j = O or No + 1. No perturbed boundary relation is required
in the longitudinal dimension where functions are assumed to be periodic. Simîlarly7 let
VyC1 and Q' be the biquadratic spline interpolants of Vn+' and Vn, respectively, defined
n f l on A2, and p, , CF+', & and c, be the interpolants of Q n f l , logdn+l, @ and log*,
respectively, defined on A3.
From Theorem 1, the above spline interpolants satisfy the time-discretized equat ions
iVA ,ive NA Ne VA NO (2-58) and (2.60) at {p.\;, ~ e ~ ) ~ = ~ , ~ = ~ : {TA,, ~e,}i,;,j,o and {qi, TO~)+;,~=,, respectively,
where
COS ,+l + (T) [(A) LI,\(;;+' + Dei/;"] = R;+l + E;+' cos 0
-n++ REf' = aQ + bVp - (CD*& + d cos 8 ~ ~ 4 ~ ) - At&,
.y = cos ë, - ($) [(A) O,\@ + *,g] cos 6
N*"ve we note that the second-order resid- To derive an expression for € y f 1 at { t l i , T ~ ) ~ = ~ , ~ = ~ ,
ual terms in EY+' arise from terms associated with the derivatives of the interpolant
namely, ($)cD,~&+' and (%Id cos B D ~ + ; + ~ . PVe first consider the former. Since
Ar* .lV$ equation (5.35) is collocated at {i,;, roJ )i=,,j=l, which align in the A-dimension with the
gridpoints of A3, the partition on ivhich 4yC' is defined, the second-order residual term
associated with ($&)cD~#;+' is given by (5.11) as - ($$) % C D X ~ ~ + ' . On the other
!VA fve hand, the collocation points {+Ai, Te, )i=i,j=l align with the midpoints of A3 in the 8-
dimension. Therefore, the second-order residual term associated with (&)d cos B D ~ P : + ~ -
is given by (5.6) as (s) COS ûD84Pf1. Combining these results: we obtain that at
VA ,ff6 {'A* 7 7sJ }i=l,j=l
AX2 I 3 --Tl 402 -." = - (5) [-- (D:Untl + DAO ) + -ï, (D:V"+' + D ; v n ) ] + O ( h r ) L4 L 12 cos 6 -
(5 -40)
€Y+' takes on a slightly different form
because îe, and 6 are gridpoints of ne (midpoints of As) while {T4)2; ' are midpoints Ne
of Â6 (gridpoints of h e ) According to Theorem 1, this irnplies that the coefficients of
l n f l the perturbation terrns associated with D@(ql + &) are different in the two cases.
The derivatives of the interpolants of the geopotential q P 1 and its logarithm satisfy
at the poles, for i = 1, - - - , iV,,, since Dec$"+'(~i\~: ka/?) = O and Dein+'(rx,, f a/2) = 0.
In order to derive a method with optimal convergence, the second-order terms in the
residuals of (5.38) to (5.41) and the boundary residuals (5.42) and (5.43) need to be
eliminated. The idea is to approximate the third derivat ives to second-order accuracy
with the spline interpolants, and then use these approximations to eliminate the second-
order terms. To this end, we extend the definitions of the difference operators AB to
handle collocation points n e z the boundâries. Along the longitude, where the boundary
conditions are periodic, the centered difference operator A,, can be redefined to be A:,
given by
for j = 1, - , ?le, while in the latitudinal direction, we combine relations (5.16) to (S.20)
to get
for i = 1, - , NA. The operator A: is useful for approximating the residuals on As, mhere
the collocation points are midpoints. We define another difference operator & for Ae,
where the first and last collocation points are gridpoints
for i = 1, - - , iV-\-
From the results in Theorem 2, the third derimtives of un+', for instance, can be
approximated to second order using the first derivative of its interpolant CI;+' and the
above difference operators. More specificaily, DSUnf' and DPUn+l can be approximated
at the collocation points of hl using DIU;+' and DeU;+', respectively? by
for i = 1, - - - , iVMi and j = 1, - , No. On the other hand, the difference operator & and
the interpolant \yCL can be used to approximate DiVnC1 at the collocation points of A2
for i = 1, , .!!\ and j = 0, . . , &. Similar relations hold for the other functions,
leading to the following fourth-order approximations for the residuals of (5.38) to (5.40)
at the collocation points of 4': A2 and hg, respectively:
for i = 1, - - , iV.1, j = 0: ive (5.46)
for i = 1 , - - - ;Na\, j = 1 ; - - - ,i& (5.47)
The points Fe, and Pol are defined as the Iatitudinal positioms ot tn of Buid particles
whose trajectories have latitudinal positions of qJ and Fe,, respectively, at tn+'. The
above equations allow us to eliminate the second-order terrns o f the residuals in (3.36) to
(5.37) and to obtain higher-order solutions, as will be explained in the sections to follom.
In the one-step method, the second-order perturbation terms imside the square brackets
in (5.44) to (5.4'7) are combined with the associated derivative operators to form perturbed
operators, which are then incorporated into the system to y ie ld a higher-order approx-
imation. For example, the first perturbation term inside the square brackets of (5.44):
(%)chi\ (D,~$;+'), cornes frorn the second term on the left side o f (5 .35) , cD,\d;". These
two terms can therefore be combined to form
where 1 denotes the identity operator. This motivates us to define the perturbed operator
Similady, the &st perturbation term inside the square brackets of (5.47), (s) ~ A ~ \ ( D , \ U ; + ' )
cornes from the second term on the left side of (5.371, --&D,\U;'L. Combining these two
terms gives
AX2 b + a n + i D*U;+l G - -0; 4
cos 8
The arrow indicates t hat D;+ cornputes the perturbed derivat ive values at collocation
points in A,\ from spline coefficients defined in A,\, or vice versa. Furthemore, com-
bining the term -, A * 2 d ~ ~ ~ B ~ ; ( ~ s + ; f L ) of (5.44) with ~ c o s ~ D ~ # ~ " of (5.35): the terni
-$di\~(D,\q!$+ l ) of (5.45) with dD,\+;+L of (5.36), and the term $R~(D~v;+') of
(5.47) with DsVF+L of (5.371, me obtain
Thus, we define the following perturbed operators:
The perturbed operator D?+', which cornputes first derivative values in the 9 direction
and the associated cesidual terrns on the Âo-gid, needs to be defuieci carefully becâuse,
as noted before, the residual terms for the collocation points { ( T , ~ ~ , î , ) (TI ; , FeNe )}:zl are slightly different from the rest. At {T,\; , î o J ) ~ ~ ; ~ ~ ~ L , the residual term associated
- * with c cos ~D*+Y+' is $c cos 9A&+' in (5.45), whereas at {(ni, Fe,), (rAi, ?O, )}zL the
ccos #&q5;+' in (5.46). Therefore, we define D;A+" residual term takes the form -= - N A Aïe- 1
at { ~ \ i ) ~ e , ) i = ; , ~ = i as
for j = 0, No.
Having defined the perturbed operators, we can derive the one-step OQSC method.
1 n+l In the one-step method, we determine the approximations Li:", Vg+' and 9, such
that they satisfy the equations
+ d cos #D;~F') 2 R2
=COS - ($) [(A) DPD; + D ~ ~ - + A v ; . ] cos 6 IV, Ne
at {?\i, {T.\~ 5% ) ~ 2 ~ ~ ~ ~ 0 and {T,\~, respectively. The approxirnate
solutions are assumed to be periodic in the longitudinal dimension and subject to the
following boundary conditions in the latitudinal dimension:
Following the procedures outlined in Section 3.6, a Helmholtz equation is derived
by eliminating the divergence terms from (5.50) using (5.48) and (5.49), and is then
linearized and solved using fixed-point iteration.
In Chapter 6, we report results of numerical experiments that show that the solu-
tions of the resulting Helmholtz equation have fourth-order errors at midpoints, which is
optimal for quadratic splines.
5.2.2 The Two-Step Optimal Quadratic Spline Collocation Method
Alternatively, the two-step optimal collocation method can be used. First : the approxi- u
mate solution, denoted by (r+', PL+', and &+', that satisfies the equations
+ (F) [(A) D~Ü;" + cos O(, cos 8
respectively, is determined. The
At 1 Rz+' = cos^^ - (y) [(-) COS 8 Ll,,Ü; + D~V;]
The approsimate functions are assumed to be periodic in the longitudinal dimension,
and subject to the following boundary conditions in the latitudinal dimension:
for i = 1: - , xi. These low-order solutions are then used to approximate the ïesidual
'n+l terrns to fourth-order. To this end, we define €2": E>+' and E~ at the associated
collocation points as
for i = 1, - - , fv,~, j = 1, - - , lVe
for i = 1, - - - , Na\, j = 0, Ne
for i = 1 , - - - ,Nd\, j = 1 , - - , N o
Similarly, the boundary residuals can also be estimated to fourth-order accuracy using
the second-order solution. So this end, we define
where (q, 1, m) = (0, 2,3) or (IV@ + 1, Ne - 1, Ne - %), and i = 1, - - ; iV,\. The terms E?+', E>+ , and E?+' are added to the right sides of (3.51) to (5.53), and
the equations (5.63) to (3.65) are solved:
cos + + ( ) [(A) D,\u;+' + DeIfif'] = R;+' + COS d 4
lV, Ne a t ~6~}i>i:z~, {TA, rd, )i,;,j,o and {rA,, 76, }zify#17 respectively, with periodic con-
ditions in the longitudinal dimension and the following boundary conditions in the O-
dimension:
for i = I, - - , ni,, to yield solution that is, as in the case of one-step OQSC, Iocally
fourth-order a t gridpoints and miclpoints and globally third-order.
5.3 Existence and Uniqueness
In this section, Ive show that the biquadratic spline collocation approximation for a
sirnplified version of the linearized Helmholtz equation (3.33) exists and is unique. As in
Section 4.3, the analysis is done on the linearized shallow mater equations with constant
coefficients, (2.12) to (9.14): subject to biperiodic boundary conditions.
5.3.1 The Unstaggered Case
We first demonstrate the existence of a unique solution for the two-step OQSC method
for the simplified shailow water equations (2.12) to (2.14) on an unstaggered grid. The
Linearized Helmholtz equation (3.33), derived by first discretizing the shallow water equa-
tions, and then eliminating the divergence terms from the discretized continuity equat ion
corresponding to (Z.l4), t akes the form
where RH is h o w n . The linear operator Ir' satisfies
which is the sarne as the operator in (4.2). Again, we prove the existence of a unique
solution for (5.66) by showing that al1 eigenvalues of are positive. The matrices in
(5.67) are defined in Appendix A. An important property of these matrices is that P
and P., share the same eigenvectors, as do Q and Qe. The eigenvalues associated with
the k-th eigenvector of P and P.! are
for k = 0, - - - : !Va\ - 1, where z E J-1. The derivation can be found in Appendix A. The
eigenvalues associated with the Z-th eigenvector of Q and Qs are
for I = 0, - - , lVe - 1.
With biperiodic boundary conditions, the second and third terms inside the square
brackets in (5.67) cancel out. Therefore, the eigenvalues of Ii, denoted by C~C, can be
expressed in terms of the eigenvalues of P, P.i7 Q and Qs as follows:
1 ( ~ 2 sin (y) cos (y)) [I. - 1 sin2 (y)] 2R cos 0=AX2 1 2 kAX 1
2 2 cos @- (12 sin (y) cos (y)) [l - 4 sin ( , )] +- Lw2 2 yJ 1 (5.68) 1 - + s i n ( )
If we let
kAX 1A8 a. = sin2 (?) . a; = sin2 (-)
then the above relation can be rewritten as
+ [ 1 - ~ k ) (1 - 2) COS OX LI;(l - a;) (1 - a) cos O X A P Q k +- A02 Q I l - a 1
Note that for aU values of k and 6
1 1 a k < l and - < 1 - - 0 5 a k 7 a ; < l + , < l - - - 5' I
3 - - 4 < 1 3 - u
Since W > O, c > O and cos 0' > O for 0' E (-ii/2,ir/2), we conclude that ok , l 2 $,
whence Ii-' exists. Consequently, a unique solution exists for the discrete Helmholtz
equation (5.66).
If the Helmholtz equation is derived analytically, the second derivatives matrices
and Qss appear in the Helmholtz operator KH, defuied in (4.4). The eigenvalues
âssociated with the k-th and 1-th eigenvectors of P.\,, and Qee are
respectively. The eigenvalues ofl of riH satisfy
W 1 , = pkql- ( G ) ' a X ~ [- COS 0% P A A , ~ ~ f cos @'pkqeel 1
Folloiving similar reasoning as for c k , l , it can be shown that a& 2 4. Moreover, since
<P.' > O, c > 0, ( 1 - a k ) / ( l - y ) 5 1
However, as will be shown in Chapter
as it tends to damp the Rossby waves.
For the one-step method, we define
and ( 1 - aj)/(l - 2) 5 1, we have o k , ~ $ O&.
'7, this alternative approach is not recommended
the following perturbed operators:
where Alg is redefined to be sirnilar to A', because of the biperiodic boundary conditions.
The operator K becomes
Again, with biperiodic boundary conditions, the second and third terms inside the
square brackets of in the above expression cancel out. The matrix PA{ shares the same
eigenvectors as P, with the eigenvalue associated with the k-th eigenvector for Pi being
respectively. Similarly, the eigenvalue associated wit h the 1-th eigenvector of Qé is
respectively. Expressed in terms of the above eigenvalues, the eigenvalues for K f take the
following form:
+ ($)2@xc [ COS exAX2 1 a i ( l - ~ l ~ ) ( ~ + y ) ~ ( ~ - d ) 2 2
Following the same logic as we did for the two-step method, we conclude that > $, and consequentiy that K I - ' exists.
5.3.2 The C-grid
When the shallow water equations are discretized on the C-grid, the Lnear operator K
appearing in (5.66) satisfies
involves the staggering operators PA-", P,?" etc.
The staggenng operators PA-" and P,:*' sharo the sarne eigenvectors as P and Px,
as do P'+A and P,-+". The eigenvalues associated with the k-th eigenvectors of pA+'
and P'+A are ezkYpk and e-'*&, respectively, where
Similady, the eigenvalues associated wit h the k t h eigenvectors of PX-'' and P , ~ + A are
-&"x ea* pz and e 2 p z , respect ively, where
In the latitudinal dimension, Q A + ~ and Q:+' share the same eigenvectors as Q and Qs,
as do Q ~ + ~ and Q:-+~. The eigenvalues associated with the 1-th eigenvectors of QL+'
and Q ~ + ~ are e t 1 Y q ~ and e - ' ' y q 7 , respectively, where
Similady, the eigenvalues associated with the I-eigenvectors of Q : + ~ and Q ~ + A are
e z L y q c and e - z ' ~ g ~ , respectively, where
The derivation of the above expressions can be found in Appendix A. The eigenvalues
of K are
1 pC2q1 ip' c -- Ck.1 = m q l - ( & q 2 { [ cos O' pl; + cos 0'-
QI pkqc2 1
4 1 -an. (1 - 2) (ip {. [- + cos O' (1 - 2) &QI
+ :?R coso- (1-7) (1 - 2) 1 kAX 2 A . 2 IAi? 1A8
cos (2-) sin (2-) sin (-i-) COS ( 2 ) (1 - 7)
2 kA \ k a \ 2 - sin (y ) COS (y) COS (-2) 3 sin - a x (5-75)
(1 - g)
where
4c al; (1 - $) + d sin(kAX) sin(lA9) - AXA0
] (5 .78)
We study the behaviour of an.,/ for typical parameter values. Since O 5 crk, a; 5 1 ,
the first term in (5.17), (1 - 2)(1- $), is always greater than or equal to a. Therefox,
D>I.,I is positive if Al, A2 > -;. Assume that At = AX = A0 h. We first consider
Al. Note that the first term inside the square brackets of (5.78) is always non-negative.
Since d = y c , the second term is rnuch srnaller than the first for s&ciently large a k
and sdcient ly smaU h. When a k 0, the first term approaches zero. Wonetheless, at
a k = O 7
so the second term also vanishes. Hence, .Al = 0. Also, for srnall positive k's, we have
LY>I- = sin2(kAX/2) FZ (kh/2) ' . Substituting this approximation into the expression for
h i Ai, and letting d = TC yields
Since cos 0% > O? the first term in the curly brackets is always positive, and since
sin(lh6) 2 -1, we have
Since f < 1, h < 1, R » b' and c z 1, we have Al = 0 ( h 2 ) . Hence? when ak FZ ( / ~ h / 2 ) ~
for a small positive integer li, -41 > -i for al! sufficiently smaU h.
Similar reasoning may be applied to AÎ to show that -A2 > -; for srna111 2 O and al1
sufficiently srna11 h. Hence or-,[ > O in the cases that we have considered. Unfortunately,
as in Section 4.32, this does not cover al1 possible cases, so we have not actually proved
( T ~ J > O for al1 k and 1. However, our numerical results in Section 6 indicate that (5.66)
has a unique solution when the simplified problem (2.12) to (2.14) is discretized on the
C-grid.
Chapter 6
Numerical Result s
Since no analytical solution is known for the general form of the two dimensional shallow
water equations (2.9) to (2.11), we introduce into the equations forcing terms constructed
in such a way that an analytical solution is known a priori. Our modified shallow water
equat ions are
dV cos 8 sin 0 CT2 + V2 -+fU+,,Qe+- dt cos2 0 R2 = Fu
d cor^-logo+ [,+hl = F+
dt cos 8
where Fu, Fu and F' are known functions of A, 0 and t , defined to yield the following
solution to equations (6.1) to (6.3):
VA, 0, t , = 2 [I - exp (-&)] cos 0 x
These functions are designed to resemble the true solutions of the original shallow water
equations (2.9) to (3.11). In our experiments, we chose the following values for the
constants
where g = 9.50616 r n ~ - ~ is the gravitational constant.
The systern was integrated on the sphere {(A, 6) E [O, 2ii] x [ -n /2 , n/2]), with
timesteps of 60 seconds (i.e. At = 60 s). The solution for 4 was computed at T = 16;
hours and compared to the reference solution (6.4), illustrated in Figure 6.1.
Figure 6.1: Reference solution for the pressure field 4 at t = 163 hours.
We solved the system using the linear Galerliin (LSG) method, the QSG method, the
standard QSC method, and the one-step and two-step OQSC methods with grid sizes
32x32, 64x64, 12Sx12S and 256x256. The Arakawa C-grid was used in our discretiza-
tion. The convergence results are shown in Tables 6.1 to 6.3 and Figure 6.2; where N
is the number of subintervals in one dimension. The error is computed as the % n o m of
the difference between the numerical solution and the reference solution at gridpoints,
divided by the 2-nom of the reference solution at gridpoints. The errors reflect the lo-
cal truncation error of each method, which, in our experiments, is found to be roughly
second-order for LSG and QSC, and roughly fourth-order for QSG and OQSC, as ex-
pected. CVe estimate the convergence order of the errors by
where el and ez are errors corresponding to grid sizes Nl and N2, respectively. The
computed values for p axe listed in the columns labelled order p" in Tables 6.1 to 6.3.
II N I QSG 1 LSG
Table 6.1: The convergence results and computational costs for QSG and LSG with
difTerent grid s izes .
L.
-
Pie rneasured the number of floating-point operations (flops) for the five spatial dis-
cretization methods using MATLAB; the results are shown in Tables 6.1 to 6.3. We
estimate the order with which the number of flops increases by
32
63
128
256
error
2.4S5e-1
1.351e-1
error
1.S36e-1
1.S6Se-2
1.471e-3
1.06Se-4
order p
N/A
3.297 - --
3.66'7
3.783
order k
N/A
1.935
order p
N/A
1.830
1.941
1.929
flops
4.947e10
1.S92e11
0ops
3.952e11
1.506e13
5.745e13
ZLl7lel3
'7.263ell
2.765el2
7.474e-2
3.S49e-2
order k
N/A
1.930 --
1.932
1.918
1.807
1.942
Table 6.2: The convergence results and computational costs for one-step and two-step
OQS C wit h different grid sizes.
3
1V standard QSC
error order p flops
Table 6.3: The convergence results and computational
iV
32
64
128
256
costs for standard QSC with
different grid sizes.
one-step OQSC
where fi and f2 are the number of flops corresponding to grid sizes NI and N2, re-
spectively. The cornputed values for k are listed in the columns labelled "order k" in
Tables 6.1 to 6.3. The conjugate gradient (CG) iterative method was used to solve the
linear Helmholtz equation (3.33), and the matrices are stored in sparse format. Thus,
the computational cost per CG iteration should increase by a factor of four when N,
the number of subintervals in one dimension, is doubled. Therefore, if the number of
CG iterations were to stay about the same as N increases, the total work in the method
should increase by about a factor of four. In our experiments, however, the flop counts
showed sub-quadratic increase, as slightly fewer CG iterations were required as N in-
creased. This, we believe, is caused by a small decrease in the condition number of the
emor
1.46Se-1
1-62-je-3
1.13Se-3
S.631e-5
two-step OQSC
error
1.651e-1
1.359e-2
1.003e-3
7.217e-5
order k
N/A
1.931
1.930
1.923
order p
N/A
3.1'75
3.836
3.730
order p
NIA
3.603
3.760
3-79,
flops
3.234e11
1 -250e1'2
4.831e12
1.84'2el3
flops
3.519el1
1.41Se12
5,402e12
3.049e13
order II-
N/A
1.951
1.947
1.934 -
-. -- ---- -- - --=-i--= - .,- - LSG
----. - standard QS? -->
Figure 6.2: A log-log scale plot of the errors versus !V (the number of sub-intervals in
each dimension) for different met hods.
matrix associated with the linear Helmholtz equation when the grid size increases. A
detailed analysis of the behaviour of the eigenvalues of the system is beyond the scope
of this thesis. The QSG method is more expensive than the OQSC methods because
the former involves computing inner products using quadrature rules, an expense not
required by the collocation met hods.
Figure 6.3 compares the efficiencies of the methods. Although the fourth-order meth-
ods require more work than the second-order methods for a given grid size; they are more
efficient for large systems ( N > 50 in our problem). This is because the cost for al1 meth-
ods considered increases at about the same rate, but the accuracy for the fourth-order
methods increases much more rapidly than for the second-order methods. Among the
fourth-order methods, the two-step OQSC method is the most efficient for our problem.
\ LSG -- 1- \\
standard QSC .-& k-
/ ',\ one-step OQSC
\ -.
cost (flops)
Figure 6.3: A log-log scale plot of the errors vers:is cornputational costs, measured as the
total number of flops, for different methods.
Chapter 7
Rossby Wave Stability
7.1 Linear S t ability Analysis
In this section, we perform a stability analysis for the quadratic spline Galerkin and
coIlocation rnethods applied to the simplified shallow water equat ions (2.12) to (2.14).
We examine the conditions under which the discretized solutions are stable. We as-
sume, for simplicity, an unstaggered grid and biperiodic boundary conditions. Staniforth
and Mitchell [30] showed that if Cartesian coordinates are used, f = O is assurned and
biperiodic boundary conditions are applied, then the Helmholtz equation should be de-
rived algebraically rather than analytically in order for the standard spatial discretization
methods tû be neutrally stable for the Rossby waves. We extend their study and demon-
strate similar results in spherical coordinates with a Coriolis parameter f that is not
necessarily zero.
7.1.1 The Continuous Case
We first examine the continuous problem. Assume that the solution to (2.12) to (2.14)
is of the forrn
where E and 1 are the longitudinal and latitudinal wave numbers, respectively, and w
denotes the wave frequency. The Lagrangian frequency u is defined by
To compute the Lagrangian frequenc~ of the tme solution, (7.1) is substituted into
the simplified shallow water equations (2.12) to (3.14). This results in a 3 x 3 system
of equations for uo , vo and 40. Then, by setting the determinant of the system to zero
and applying (7.2), an expression for the Lagrangian frequency v can be obtained. This
expression can then be solved to yield the solutions
k u = 0 7 * J w [( R cos 0- 1 2 + ($)1 +p2
The first solution, u = O or w = -u=/(Rcos 0') -v'/R, corresponds to the solution of
interest - the Rossby waves. The other two frequencies are associated with the gravity
waves.
7.1 -2 Algebraic Derivation of the Helmholtz Equation
We now solve the simplified shallow water equations (2.12) to (2.14) using the two-
level semi-Lagrangian semi-implicit time integration method with a finite-element spatial
discretization scheme. We analyze the stability of the solution when the Helmholtz
equation is derived algebraicdly; in other words, we compare (7.3) to the Lagrangian
frequency of the discretized solution when the divergence terms are eliminated after
spatial discretization. We assume that the longitudinal mass matrix P and the first
derivative matrix P,\ associated with the finite-element scheme have common eigenvectors
g k = {,zkjAA LVA-L } for k = 0, - - - , Nd\ - 1. Let the associated eigenvalues be p i and p.\,,
respectively- Thus we have
Similarly-, we assume that both the latitudinal mass matrix Q and the first derivative
matrix Qs have cornrnon eigenvectors denoted by 19; = {e"jAB )J=o for I = 0, - - - , No - 1,
with the associated eigenvalues denoted by ql and qdi, respectively. Thus we have
The discretized solution a t time-level t,+l is assumed to be of the following wave form:
QSG, QSC and One-Step OQSC Methods
The mass and first derivative matrices arising frorn the quadratic spline Galerkin (QSG):
the quadratic spline collocation (QSC), and the optimal quadratic spline collocation
(O QSC) methods with biperiodic boundary conditions satisfy relations (7.4) and (7.5),
with the eigenvalues for each method as indicated in Appendix A. In this subsection,
we study the stability of the Rossby solutions obtained with the QSGI the QSC and the
onestep OQSC methods. Analysis for the two-step OQSC method follows in the next
subsection.
For an arbitrary function +(A, 8, t ) , let $"(A, 8) $(A-u'At/(Rcos Qu), O-vR4t/R, t,).
Discretizing the simplified shallow water equations (2.12) to (2.14) in time using the semi-
Lagrangian semi-implicit ( S LSI) met hod yields
Following the notation introduced in Chapter 3, the spatially discretized form of equation
(7.7) is
1 + R cos 0%
In order to facilitate the eigenvalue analysis, we rewrite equation (7.10) in terms of the
target solution U A , V A and +a, instead of the spline coefficients Un+ ', Vn+l and ORf ',
since the target solution is assumed to be of the wave form (7.6):
With biperiodic boundary conditions, al1 matrices in ~arentheses in (7.11) cornmute. By
pre-multiplying (7.11) through by (E\ Q Es) and simplifying we obtain
Following a similar procedure, cve
(7.9):
obtain the following discretized form of (7.8) and
When the discrete solution ('7.6) is evaluated a t the midpoints of the partitions,
the resulting set of values equals a scalar multiple of an eigenvector of the discretized
operators. To illustrate this: let dkVl 79k @ 91 = ( e ~ ( ~ ~ . \ + l j A Q ) 'Vx-111v6-1 i=ûj=û , for k =
0,. - - , NA - 1 and I = 0: - - - : - 1: be an eigenvector of the discretized operators.
When evaluated at midpoints, the geopotential solution 4:+' takes the form
1 6ki1 ((i + =-)Ah. - ( j + L)Ao) 2 = he'[k~i+~~4*+l(j+~)~~+~(n+1)~t]
operators. These eigenvectors are t hen substituted into equations (7.12) to (7.14). Note
also t hat
and that sirnilar relations hold for TAY' and v;+'. Using relations (7.1) and (7.5), we
obtain
To simplify notation, we have dropped the subscripts k and 1 in the eigenvalues ps, p ~ k ,
ql and q e l . The above equations hold for k = 0, - - , Na, - 1 and I = 0, - - - : 1% - 1.
Equations ('7.15) to (7.17) form a 3 x3 system for uo , vo and 40. By setting the
determinant of the resulting system to zero to find a nontrivial solution to (7.13) to
(7-17), an equation for ezYAt c m be obtained. The roots are found to be
where
Since eZuAt = 1, or v = O, is one of the solutions, the numerical scheme is neutrally stable
for the Rossby waves. The solutions for the gravity waves are also bounded since
Therefore, the numerical scheme is also stable for the gravitational modes. The accuracy
of the gravity waves will be considered in Chapter S.
Two-Step OQSC Method
In the two-step OQSC method, a second-order intermediate solution is cornputed first,
and is then used to perturb the equations appropriately. The perturbed equations are
then solved to yield a locally fourth-order solution (see Section 3.2.2 for details). The
mass and first derivative matrices for the two-step OQSC method are the same as those
for the QSC method. In fact, the solution obtained from the first step of the two-step
method, tvhen the unperturbed equations are solved, is that of the QSC method. We
denote this intermediate solution by
The Lagrmgian frequency ü of the intermediate solution can be computed following the
steps described in the previous section; and is given by (7.18); mith the eigenvalues p;
q, p,! and q6 given in Section A.%. In the second step, because of these perturbation
terms, the associated 3 x 3 system for uo, vo and do is different from that derived in the
previous subsection. We assume that the solution of the second step satisfies (7.6). Note
that the Lagrangian frequency ü of the intermediate solution may be different from the
Lagrangian frequency u of the solution of the second step. Since the equations solved in
the second step of the two-step collocation method are perturbed, the eigenvalue analysis
for this method needs to be done differently. Using relations (5.6) and (5.7), and following
procedures similar to t hose outlined in Section 5.2, we obt ain the following discretized
perturbed equations for the ..\-grid:
For the two-step OQSC method, the centered difference operators AA and A8 are the
same as the stifiess matrices Pxx and Qee, respective'v, and thus they have the same
eigenvalues pal* and qee, respectively, which can be found in Appendix A. Rewriting
equations (7.20) to (7.22) in terms of the eigenvalues and simplifying, we obtain
pq (ew:-- 1) u o - Y P ~ (eau: + 1) + R cos 0.1
(7-25)
By setting the determinant of the above system to zero, and solving for eZuAt, we find
where
Since etYnt = 1, or v = O, is one of the roots, the numericat solution for the two-step
OQSC method is neutrally stable for the Rossby waves. We have also verified using
MAPLE that the gravity solutions are bounded.
7.1.3 Analytic Derivation of the Helmholtz Equation
A genuine Helmholtz equation may be derived by eliminating the wind velocity depen-
dence from the continuity equation (2.14) before spatial discretization. T h e simplified
shallow water equations (2.12) to (2.14) are fist discretized in time to yield
Un+l - ,ijn 1 At ) + R cos R-
= O
p + l - At 2 R cos 0'
We solve (7.27) and (7.28) for un+' and un+' in terms of $yf1 and PiC' as we did in
Section 2.6. Then taking t h e A- and 8-derivatives of t he resulting equations, we get
which can be considered as a special case of (2.61) with constant Coriolis parameter.
Equations (7.30) and (7.31) can then be used to eliminate the unknown divergence terms
from the continuity equation (7.29) to yield
b + a- (g) [(s) 6: - b q + - 6; + (1 + a)GS] = 0 cos O=
Recall that Pu and Qse denote the stifiess matrices in the A- and &directions,
respectively, and p * ~ , and qgs, are the eigenvalues associated with their eigenvectors f i k
and d i , respectively, as d e h e d in Section 7 - 1 2 The spatialiy discretized form of equation
(7.32) is
1
- (t) 2R c @. 4 Q ) + ( P @ Q e e ) (CA 8 &)-L (&+l + 4,) 1
+ m- (g) {[ ' t a (Pi g~ Q ) - b ( P @ Q e ) (EA @ Ed-% 2R cos 01 1
b
+ [ (PA 8 Q ) + ( 1 + a ) ( p 8 QO)] (EA O EO)-'G;) = 0 cos Of
(7.33)
where we have writ ten (7.33) in terrns of the solution values #:+' rather than the spline
coefficients 8"+ ', as we did earlier in transforming equation (7.10) to (7.11). Recall
that the matrices in parentheses commute given biperiodic boundary conditions. By
pre-multiplying (7.33) by ( E A 8 E o ) and simplifying, we obtain
l + a [-(Pd\ cos O= @ Q ) - b ( P @ Q,)] Gl,
b [-(PA COS O= @ Q ) + (1 + a ) ( P @I Q O ) ] GE) = O
(7.34)
Substituting the solution (7.6) into the above equation, rewriting the resulting equation
in terrns of the eigenvalues and simplifying, we obtain
By setting the determinant of equations (7.13), (7.16) and (7.35) to zero, an equa-
tion for is obtained. It is found that et"*' = 1 is no longer a solution to this
frequency equation, which implies that an undesirable phase shift haç been introduced
into the Rossby mode. Therefore, in our form~dation, the Helmholtz equation 3s derived
algebraically to preserve the neutral stability of the Rossby waves.
7.2 The Equatorial Rossby Wave
The equatorial solitary wave described by Boyd [4] is frequently used to test t h e stability
of a numerical method for the shallow water equations with respect to Rossby waves. To
use this test problem on Our model, we transform Boyd's formulation, originally written
in Cartesian coordinates, to spherical coordinates. We adopt Boyd's notation and let 4'
be the geopotential perturbation; that is, if we denote the reference geopotential by the
constant @, then C$ = 6 + #. The equations are nondimensionalized using the length
and time scales
where E = 4R2R21g& is the Lamb parameter. In our implementation, R is nondi-
mensionalized by L, whereas the longitude X and latitude 8 are left unchanged; Le.,
O 5 X 5 2ïr and - r / 2 5 0 $ ii/2. The resulting nondimensional nonlineu shailow water
equations on the equatorial ,&planeL are
du -- R8v + 4 dt
= O R cos 8
du 4 = 0 - + RBu + - d t R
d 1 - log(1 + 4') + dt
[ux + (v cos = O R cos 8
The @-plane approximation, valid for fluid motions restricted to a srnaIl range of latitudes, assumes that the Coriolis force varies linearly in the north-south direction [l]. For motions in close prosirnity to the equator, the Coriolis parameter is approximated by R0.
The above equations are integrated with the following initial conditions
where
In our study, a timestep of 0.05 non-dimensional units (At = 0.05, approximately two
hours) is used. Experiments are conducted ivith grids of sizes iV = 32 x 32, 64 x 64 and
12s x 12s. Variables are staggered using the C-grid.
The equations descri be an equatorial soliton which slowly propagates westward, with
no change in shape. Figure 7.1 shows the result of a long simulation, using the two-step
OQSC method on a 1% x 1'18 grid for 24 non-dimensional time units, approximately 41
days. Similar results were also obtained using the QSG and one-step OQSC methods. In
our experiments? the height fields lost about 5% of their initial arnplit ude which traveled
eastward as equatorial Kelvin waves. We believe this is because the initial conditions
(7.39) to (7.41) are inexact, as suggested by Iskandarani et al. pl] and M a [[22]. The soli-
ton propagated westward, as predicted, with little change in amplitude or phase, t hereby
confirming our analysis that spatial discretization with quadratic spline collocation is
neutrally stable for the Rossby modes. The small arnount of dispersion present can be
exp lained as follows. \Vit h semi-Lagrangian integration, variables are needed at depar-
tures points, which are usually off-mesh points. Cubic Lagrange interpolation is used
to estimate function values at departure points. Spatial interpolation causes damping
and phase shift. Compared to linear or quadratic interpolation, the eRects introduced by
cubic interpolation are less severe. The finer the mesh, the less prominent the damping
and dispersive effects should be [2, 241. This is confirrned in our experinents (compare
Figure 7.1 wit h Figure 7-21.
Figure 7.1: A simulation by the two-step OQSC method of a Rossby soliton traveling in
the direction of decreasing X values on a 128 x 125 grid for 24 time units (approximately
41 days).
Figure 7.2: A simulation by the two-step OQSC method of a Rossby soliton traveling
in the direction of decreasing X values on a 32 x 32 grid for 24 time units. Substantid
dispersion, caused by spatial interpolation, can be observed.
Chapter 8
Cornparison of Staggering Schemes
In this section, we analyze the performance of three staggering schemes, namely the
Arakawa A-grid, B-grid and C-grid, which are depicted in Figure S-1. We concentrate
on the group velocities of gravity waves, the accuracy of which is sensitive to the choice
of discretization scheme. Our analysis of the simplified shallow water equations (2.12) to
(2.14) with biperiodic boundary conditions shows that care must be taken in choosing
an appropriate discretization scheme, otherwise energy associated with short gravity
waves may propagate in the wrong direction. This conclusion is supported by numerical
experirnents involving equatorial Kelvin waves.
8.1 Linear Stability Analysis
A stability analysis is performed on the quadratic spline Galerkin and collocation meth-
ods applied to the simplified shallow water equations (2.11) to (2.14) with biperiodic
boundary conditions. Our focus is on the group velocities of gravitational modes. Group
velocity can be computed from wave frequency, as will be described in the next subsec-
t ion.
Our malysis is done on a normalized system with P = 1, R = 1, 0' = O and = 0,
with the additional assumption that the mean wind speeds are zero; i.e. u' = v* = 0.
4 P Y . L A
Figure 8.1: A schematic diagram of the Arakawa A-, B- and C-grids.
The discretized solutions are computed with At = 0.1, AX = 0.25 and A8 = 0.25.
81.1 The Continuous Case
The wave frequency of the gravity solution of the continuous problem can be derived
from (7.2) and (7.3) as
u= v* Wf = V - --
R cos O- R
- - IL= -
R cos O= R R cos O=
The group velocities of the gravity waves in the A-dimension are given by
I dw*
while in the 0-dimension the group velocities are given by
dw* cg+ 2 -
dl
In this section, we will focus on the positive components of the group velocities c:
and ca. The group velocities for the continuous solution are shown in Figures 8.2 and
8.3. Note that c: 2 O, c r 2 O for a11 wavelengths.
Figure 8.2: Group velocities of continuous gravity solution in the A-direction. Note that
aU waves travel in the positive direction.
8.1.2 The A-grid
The A-grid is unstaggered, and the Lagrangian frequencies of its discrete solution com-
puted with the quadratic spline Galerkin (QSG), the quadratic spline collocation (QSC),
and the one-step optimal quadratic collocation (OQSC) methods can be derived from
('7.18), since
Then, using (7.2). we can compute the wave frequencies of the numerical solution as
where A and B are defined in Section 7.1. Note both A and B Vary with k or 1 since
they are are functions of the eigenvalues p, p ~ , q and ge, which in turn depend on the
wave numbers k and 1. In addition, note that the expressions for p, p,!, q and qe depend
Figure 8.3: Group velocities of continuous gravity solution in the O-direction. Note that
d l waves travel in the positive direction.
on the numerical method used. These values are given in Appendix -4 for the QSG, QSC
and OQSC methods.
The wave frequencies of the gravity solution obtained wit h the two-step OQSC method
can be obtained from (7.26) as
p t - - 4.4 - B + C f Z Z , / ~ A ( B - 6) + (4 + f =2)(bX2 ; cos W26d2)A + D = E 4 A + B
çvhere A, B, C, D, SA and 66' are defined in Section 7 - 1 2 Note that C , D, 6X and SB
have imaginary parts.
The group velocities of gravity waves for different wave numbers corresponding to the
QSG, QSC and OQSC methods on unstaggered grids are shown in Figures 8.4 to 8.11.
Note that the group velocities of short waves (large wave numbers) are negative, even
though, according to our analysis of the continuous solutions, al1 gravity waves should
Figure 5.4: Group velocities of gravity waves in the A-direction for the QSG
an A-grid. Note that some waves travel cvith negative velocities.
method on
travel in the positive direction. It can, therefore, be concluded that the (unstaggered)
A-grid is inaccurate in capturing the group velocities of short gravity waves.
8.1.3 The B-grid
On the B-grid, the wind speed components u and v are computed at the same locations,
while the geopotential 4 is evaluated at points that are staggered with respect to the
u, v-grid in both X and 0-directions. Let G:+' and GY' denote the vectors of values of
U n + L and v:+' at the rnidpoints of the u, v-grid, and +",+' denote the vector of values of
42' at the midpoints of the &grid. N 1
( L\-L
Recall that Gx. = (ezkjA") j2,- - . Let J1; = ez"(j+4)4a' for k = O , - a - , f$ - land 1 j=o
observe that the staggering operators satisfy conditions similar to ('7.4):
Figure 8.5: Group velocities of gravity waves in the O-direction for the QSG method on
an A-grid. Some waves travel with negative velocities.
Since = e t g A X ~ k 7 ive see that is an eigenvector of PA-'' and ~,f '+' , with p2eiTA"
and eigAA being the associated eigenvdues, respectively. Sirnilarly, an- is an eigenvector
of and P,?+", ivith p2e-i54" and e- ' :~ " being the associated eigenvalues,
respectively. The arrows in p z and p z indicate the fact that these scalars are associated
with staggering operators. The scalars p z and p c take on different values depending on
the choice of spatial discretization rnethods (see Appendix A). Sirnilar relations hold for
the corresponding staggering operators in the 8-direction.
Figure 8.6: Group velocities of gravity waves in the A-direction for the QSC method on
an A-grid. Some waves travel with negative velocities.
Q S G , QSC and One-Step OQSC Methods
With the B-grid, the discretized equations (7.12), (7.13) and (7.14) become:
1 Q ~ + ~ , (4:''; 4:) = O R cos OL (8 -7)
a* R cos Ox
Figure 8.7: Group velocities of gravity waves in the 8-direction for the QSC method on
an ,4-grid. Some waves travel with negative velocities.
Following the procedures described in Section 7.1.2, we rewrite equations (S. 7) to (8.9)
in terms of the eigenvalues of P and Q, and obtain the following after some simplifications:
+ 1 R cos 0% (S.10) R
The solution for the Lagrangian frequency for the l3-grid is given by the relations
where
Figure 8.S: Group velocities of gravity waves in the A-direction for the one-step OQSC
method on an A-grid. Some waves travel with negative velocities.
The group velocities of gravity waves discretized by the QSG, QSC and one-step
OQSC methods on a B-grid are shown in Figures 5.12 to 8-17. As with the A-grid,
energy propagates in the wrong direction at smali scaies for al1 the methods considered,
though the problem is less severe in this case.
Two-Step OQSC Method
For the two-step OQSC method, the discretized perturbed equations (7.20) to (7.22) for
the B-grid take the following form:
Figure 8.9: Group velocities of gravity waves in the 0-direction for the one-step OQSC
method on an A-grid. Some waves travel with negative velocities.
(S. 14)
Figure 8.10: Group velocities of gravity waves in the A-direction for the two-step OQSC
method on an A-grid. Some waves travel with negative velocities.
Rewriting the above equations in terrns of the eigenvaiues and simplifying yields
I X 2 3 3
Rcos O= 24 PAXP.\ Q
p h f ( e z Y q + 1) R cos 0-
(S. 1s)
Figure 8.1 1: Group velocities of gravity waves in the O-direction for the two-step OQSC
method on an A-grid. Sorne tvaves travel with negative velocities.
(S. 19)
By moving all terms to the left side, setting the determinant of the resulting system
to zero, and solving for etY"'; we find that
where
Figure 8-12: Group velocities of gravity waves in the A-direction for the QSG method on
a B-grid. Some short waves propagate with negative velocities.
It hos been verified using MAPLE that (GI is bounded by 1, so the gravity solution
is bounded. The group velocities can be cornputed from the wave frequencies, which are
given as
and are shown in Figures 8-18 and 8.19. As with the QSG, QSC and one-step OQSC
methods, some short gravity waves travel in the wrong direction.
Figure 8.13: Group velocities of gravity waves in the O-direct ion for the QSG
a B-grid. Some short waves propagate with negative velocities.
8.1.4 The C-grid
rnethod on
The C-grid is used in our irnplementation of the shallow water equations; its structure has
been described in Section 3.1 and depicted in Figure 8.1. In this case, the three target
functions are evaluated at different locations, with the wind speed components u and
v staggered with respect to the geopotential 4 in the A- and 8-directions, respectively
"+' and $:+' evaluated at the Let u:+', v:+l and +y1 be vectors of values of u F L , v p
midpoints of the associated partitions, respectively.
QSG, QSC and One-Step OQSC Method
With the C-grid, the discretized equations corresponding to (7.12), (7.13) and (7.14)
become
Figure 8.14: Group velocities of gravity waves in the A-direction for the QSC method on
a B-grid. Some short waves propagate with negative velocities.
The above equations, written in terms of the eigenvalues, take the form:
$0 = O (S.25) R cos 0-
5
O O
Figure 8.15: Group velocities of gravity waves in the O-direction for the QSC method on
a B-grid. Some short tvaves propagate with negative velocities-
The solution for the Lagrangian frequency for the C-grid is given by the relations
where
A = (RCOS o = ~ ~ ) ~
2 -t2 B = (At f R cos 0'pi4i)2 - 4 t z @ = ( p ~ 2 q Z + cos2 O'p q, )
Figure 8-16: Group velocities of gravity waves in the A-direction for the one-step OQSC
method on a B-grid. Some short waves propagate with negative velocities.
Two-Step OQSC Method
For the two-step O QS C method, the discretized perturbed equations (7.20) to (7.22) for
the C-grid take the following form:
Figure 8.17: Group velocities of gravity waves in the 0-direction for the one-step OQSC
method on a B-grid. Some short waves propagate with negative velocities.
Rewriting the above equations in terms of the eigenvalues and simplifying yields
Figure 8.1s: Group velocities of gravity waves in the A-direction for the two-step OQSC
method on a B-grid. Some short waves propagate with negative velocities.
By moving al1 terms t o the left side. setting the determinant of the resulting system
to zero, and solving for ezY'", we find that
4 22 J ~ A ( B - C ) - At2@-[4A + (At f -R cos 0'p+q+)2] (bX2 + cos 0=2b02) + D
Figure 8.19: Group velocities of gravity waves in the 6-direction for the two-step OQSC
method on a B-grid. Some short waves propagate with negative velocit ies.
where
2 -9 B = (At f ' R cos 0'p+q-')2 - 4 t 2 ~ ' ( p ~ 2 q2 + cos2 O'p qo )
It lias been verified using MAPLE that (el is bounded by 1, so the the gravity
solutions are bounded. The group velocities can be computed from the wave frequencies,
which are given as
Figure 8.20: Group velocities of gravity waves in the A-direction for the QSG
a C-grid. The group velocities of al1 scaies are positive.
method on
Figures S.20 to 8.27 show the group velocities of gravity waves for the QSG, QSC
and OQSC methods, discretized on a C-grid, for different wave numbers. In this case,
the group velocities are positive for aJJ scales, which implies that energy propagates in
the proper direction. Hence, rve conclude that the Arakawa C-grid should outperforrn
the other staggering systems in that it more accurately captures the group velocities of
gravity waves of al1 wavelengths. Tt is also interesting to note that there is noticeable
retardation in the group velocities of the gravity solution for the QSG method, but not
for the QSC and OQSC methods. However, a detailed analysis of this phenomenon is
beyond the scope of this thesis.
Figure 5-31: Group velocities of gravity waves in the 6-direction for the QSG method on
a C-grid. -4s in the A-direction, group velocities of al1 scales are positive.
8.2 The Equatorial Kelvin Wave
We now demonstrate numerically the effects staggering has on group velocities of short
gravitational waves. The nonlinear equatorial Kelvin wave, investigated by Boyd [517 is
used as the test case in our study. Equations (7.36) to (7.35) are integrated with a new
set of initial conditions:
' -(i18)2/2-(RX cos B/rlX)? #'(A, 8, 0) = ~e
The above initial conditions generate equatorially trapped gravity waves, traveling
eastward, with wavelengths of the order of the longitudinal grid spacing. Results for the
two-step OQSC method are shown in Figures 8.36, 5.37 and 8.38 on A, B and C-grids
Figure 8-22 Group velocities of gravity waves in the A-direction for the QSC
a C-grid. Group velocities of al1 scales are positive.
method on
respectively. The simulations were run for 8 non-dimensional time units on a 64 x 64
grid. Similar results are also obtained using the QSG and one-step OQSC methods.
When the A or B-grid is used, short gravity waves are observed t o propagate westward,
which is the wrong direction, thus demonstrating the failure of both the A and B-grids to
faithfully capture the group velocity of short gravity waves. On the other hand, gravity
waves of a11 scales propagate in the right direction when the C-grid is used.
Figure 8.23: Group velocities of gravity waves in the 0-direction for the QSC method a
C-grid. Group velocities of al1 scales are positive.
Figure 'e-24: Group velocities of gravity waves in the A-direction for the one-step OQSC
method a C-grid. Group velocities of a11 scales are positive.
Figure 8.25: Group velocities of gravity waves in the 8-direction for the one-step OQSC
method on a C-grid. Group velocities of al1 scales are positive.
Figure 8-26: Group velocities of gravity waves in the A-direction for the two-step OQSC
method on a C-grid. Group velocities of a11 scales are positive.
Figure 8-27: Group velocities of gravity waves in the O-direction for the two-step OQSC
method on a C-grid. Group velocities of a11 scales are positive.
Figure 8.28: Group velocity errors of gravity waves in the A-direction for the QSG met hod
on a C-grid.
Figure 8-29: Group velocity errors of gravity waves in the 0-direction for the QSG method
on a C-grid.
Figure 8-30: Group velocity errors of gravity waves in the A-direction for the QSC method
on a C-grid.
Figure 8.3 1: Group velocity errors of gravity waves in the B-direction for the QS C m e t hod
on a C-grid.
Figure S.32: Group velocity errors of gravity waves in the A-direction for the one-step
OQSC method on a C-grid.
Figure 5.33: Group velocity errors of gravity waves in the 8-direction for the one-step
QSC method on a C-grid.
Figure 8-34: Group velocity errors of gravity waves in the A-direction for the two-step
OQSC method on a C-grid.
Figure 8.35: Group velocity errors of gravity waves in the 0-direction for the tcVo-step
QSC method on a C-grid.
Figure 8.36: Simulation of gravity waves propagating eastward (in the increasing X-
direction) with the two-step OQSC method on an A-grid. Some gravity waves travel in
the wrong direction.
thefa p i c ? - O ' lambda
Figure 8.37: Simulation of gravity waves propagating eastward (in the increasing X-
direction) with the two-step OQSC method on a B-grid. Again, some gravity waves
propagate in the wroag direction (in the decreasing A-direction)-
Figure S.38: Simulation of gravity waves propagating eastward (in the increasing X-
direction) with the two-step OQSC method on a C-grid. The group velocities of waves
of wavelengths have been captured correctly.
Chapter 9
Conclusions
9.1 Results
Spatial discretization schemes commonly used in meteorological applications are cur-
rently limited to spectral met hods or low-order finite-difference/finite-dement met hods.
In order to irnprove spatial accuracy, we present three algorithrns that combine the
semi-Lagrangian semi-implicit (SLSI) time integration method and high-order spatial
discretization methods for solving the shallow water equations on the sphere. Al1 three
spat in1 discretization met hods investigated in t his t hesis belong to the class of weighted
residual methods and are based on quadratic splines. The following is a summary of the
contributions of this thesis.
1. We point out that, when spatial discretization is done on staggered grids using
quadratic splines, the boundary conditions at the poles may not be given at the
end points for some partitions. In Section 3.2.4, we explain how the quadratic spline
basis functions should be adjusted to satisfy the latitudinal boundary conditions
associated with As (see Section 3.1.1), where the poles are rnidpoints instead of
gridpoints.
2. With the SLSI time discret ization method, the functions are integrated or averaged
along fluid t rajectories. In the tirne-discretized equations, functions a t time-level
t , are evaluated at upstrearn points. In Section 3-5, tve explain how such upstream
functions should be represented in the biquadratlc spline approximation space, and
how their nonhomogeneous boundary conditions should be handled.
3. W e combine the quadratic spline Galerlcin (QSG) method with the SLSI method
to solve the shallow water equations, and discuss; how the inner products should be
evaluated when spatial discretization is done on- staggered grids. We also demon-
strate mathematically, on both the unstaggered grid and the C-grid, the existence
and uniqueness of the numericd solution for the simplified shallow ivater equations
(2.12) to (2.14) with biperiodic boundary condit .ions.
4. In Chapter 5 , ive describe how the shallow ivater equations can be solved using a
combination of the SLSI method and the one-step and two-step optimal quadratic
spline collocation (OQSC) methods. Previous nqork on OQSC methods is limited
to unstaggered grids. We extend the OQSC rmethods to a system of first-order
differential equations on staggered grids. As we h a v e clone for QSG, we demonstrate
mathematically, on both the unstaggered grid a n d the C-grid, the existence of a
unique solution for the OQSC methods for the simplified shallow water equations
(2.12) to (2.14) with biperiodic boundary conditions.
5 . In Chapter 6, we present some numerical results which support Our conjecture that
the QSG and OQSC methods have a fourth-order spatial convergence rate. We
also compare these methods to the second-order linear spline Galerkin and the
s tandad (non-op timal) quadratic spline colloca-tion methods. We also compute
the computational costs of the methods on our &est problems, and conclude that,
among the methods studied, the two-step OQSC method is the most efficient, in
the sense that it achieves the smallest error for a given amount of computational
work.
6. With linear stability analysis, S taniforth and Mitchell [30] showed that if Cartesian
coordinates are used, f = O is assumed and biperiodic boundary conditions are
applied, t hen the Helmholtz equation should be derived algebraically rat her t han
analyt ically in order for the standard spatial discretization met hods to be neutrally
stable for the Rossby waves. We extend their study and demonstrate similar results
in spherical coordinates with a Coriolis parameter f that is not necessarily zero.
We then verify, using Boyd7s equatorial soliton test case: that the QSG and the
OQSC methods, when applied in conjunction with the SLSI rnethod- axe indeed
stable and non-dispersive for the Rossby waves.
7. Finally, Ive compare the performance of the QSG, QSC and OQSC methods on three
different staggered grids, first mathernatically through linear stability analysis, and
then experimentally with the Kelvin wave test problem. We find that the C-grid
offers the best accuracy for gravity waves for the new discret izat ion met hods st udied
in this thesis, confirming the general agreement that the C-grid is best for the
standard discretization methods.
9.2 Future Work
It is our plan to theoretically prove the convergence behaviour, which have been shown
numerically in this thesis, for the QSG and OQSC methods. Work is in progress for
deriving the error bounds for, as well as the existence and uniqueness of, the solution for
the O QSC method for a one-dimensional first-order system discret ized on a staggered
grid. Our next goal is to prove similar results for the two-dimensional case.
The spectral methods, which yield high-order solutions, give rise to dense matrices
and are therefore non-scalable. On the ot her hand, finiteelement met hods are readily
parallelizable. Therefore, high-order fini te element met hods seem to be a viable alter-
native to spectral methods. The techniques presented in this thesis for quadratic spline
rnethods should be extendible to higher order spline methods, assuming the extra bound-
ary conditions required by high order splines are set appropriately and the appropriate
perturbation terms for collocation are derived. It would be an interesting project to
compare the performance of spectral and high order finite element methods.
The spherical coordinate system suffers from many problems at the poles. For in-
stance, the tvind velocity components u and v take on multiple values a t the poles, To
avoid this difficulty, we compute, instead, the wind images U and V 7 which vanish at
the poles by construction. Close to the poles, however, Zi and V tend to be smail, which
may generate large errors when the values of u and v are recovered to compute the fluid
trajectories. Though this has not caused any noticeable problem in our experiments, how
best to cope with the "pole problem" should nonetheless be investigated further.
The boundary conditions on U and V at the poles are naturaIly homogeneous Dirich-
let, and those for the geopotential 6 are designed to mimic the behaviour of its spherical
harmonic expansion, for which the normal derivative vanishes at the poles. Though
these boundary conditions have been adopted in the literature, it is conceivable that the
geopotential may be changing dong the latitude at the poles (i.e. have a nonzero nor-
mal derivative). A consistent and realistic set of boundary conditions, or a novel spatial
discretization may prove to be interesting research projects.
The NCAR shallow water equations test set, developed by Williamson et. al. [36] ,
which is cornposed of seven test cases with different levels of complexity, is designed to
evaluate numerical methods proposed for weather prediction and climate modeling, and
will be used to further verify that our methods indeed perform well on these problems
in cornparison with other currently adopted numerical methods such as the spectral
transform method. Special attention will be paid to the spatial convergence behaviour
of the numerical solution, and its sensitivity, if any, at the poles.
Given their computational intensity, weatheï prediction problems are perfect candi-
dates for parallel irnplementation. Therefore, it is our goal to develop numerical methods
that are scalable on massively parallel machines. Based on finite-element schemes, our
met hods lead to sparse matrices and t herefore generate relatively Little global communi-
cation, compared to the spectral transform methods which give rise to dense matrices.
A paralle1 version of our code will be deveioped, and then tested with one of the prob-
lems in the NCAR test set that is identified aç an efficiency benchmark for assessing the
performance of parailel algorit hms.
The next logical step beyond the shallow water equations test problem is to build a
primitive equat ion dynamical core. A primitive equat ion dynamical core differs from the
shallow water model in t hat the former is three-dimensional, includes temperature and
also viscosity and hence dows turbulent flows [17]. Though the shallow ivater model in-
cludes many important characteristics present in more comprehensive atmospheric mod-
els, it is insufficient in part because it includes only deterministic cases. A primitive
equation dynamical core, on the other hand, includes turbulent cascade in tvhich dissipa-
tion is an important component. The Held-Suarez test [1S] andlor the Boer-Denis [3] test
may be used to determine the stability and accuracy of the methods, and the sensitivity
to resolution, timestep and adaptive t imesteps.
Appendix A
The Discrete Operators
In this appendix, we present the explicit forms of the quadratic spline Galerkin (QSG)
and the quadratic spline coLlocation (QSC) operators, as well as their eigenvectors and
the associated eigenvalues. Throughout , biperiodic boundary conditions are assumed.
Let z J-i Let the following vectors d c and di in the A-direction be defined by
where AX = Lïï/lVai and k = 0, - - , f i - 1. Similarly in the &direction, we define
where Ad = 2ii/.Ne and 1 = 0, - - , No - 1. Note that dk = etkA.~/211k, and similarly that
8; = ezkA6/29r ~h ese vectors are useful in deriving the eigenvalues of the linear operators
considered in this appendix.
A.1 The Quadratic Spline Galerkin Method
A. l . l The Matrices
The matrices associated with the QSG rnethod can be obtained by analytically computing
the inner products of the quadratic spline basis functions defined in Section 3.1.1. With
APPENDIX A. THE DISCRETE OPERATORS
biperiodic boundary conditions, the rnass matrices are found to be:
- for i, j = 1,..- ,Nd, and k , l = 1,s.- , N o . PVe have also piiA - (PA+')= and Q'+* =
(QA-'*)*- The first derivative matrices are defined belovc*:
A+A - - (p ,p+B)~ and Q$+A = for i: j = 1' - - , IV,, and k, 1 = 1, - - - , N e , whereas PA -
-(Q:+')~. In the above definitions, As = AX for the P-matrices and 4 6 for the
Q-matrices. Finally, the stiffness matrices have the following form:
for i, j = l , - . . ,NA and k, l = 1 , - - - ,Ne.
A.1.2 Spectral Properties
Below we give the form of the eigenvalues of P , f i and P.,,\. Note that with biperiodic
boundary conditions, both 1 9 ~ and di, defined in (A.1); are eigenvectors of P, Pd\ and
Pa\.\\. The eigenvalue ph for P corresponding to both dk and 9^k is
while that for PA is
- - A {S sin (y) cos (9) [I. - 2 sin2 (y)] + 40 sin (1) ~ A X cos (_) IZAX ) 24AX
- 2 - kAX kAX kAX 2- 3AX sin (1) cos (1) [3 - sin2 (T)]
and that for Psix is
Sirnilarly, 8; and 9; are the eigenvectors of Q, Qs and Qss , and the corresponding
eigenvalues, denoted by qi, qd, and qss,, respectively, are similar to pi, p.,, and p.,-,,,
respectively.
The eigenvectors 91 and 81 satisfy the following relations with the stagering matrices:
where
= -. kAA 2 [cos (F) + 237 cos (y) + 1682 cos (2-)]
3840
- 1 kAX k A A kAX - -cos 120 (-) 2 [120 - 60sin2 (& + sin4 (-531 (A.7)
-+ - 5kAA kAA P X , - z L AX [sin (T) + 75 sin (y) + ~ 4 s i n ( T ) ]
- - 33 kAA kAX kAX i n AX ( ) k 4 - 20 sin2 (_) + sin* (T)] (A-8)
The matrices QA*', Q ~ + ~ ? Q : + ~ and Q$+A satisfy relations similar to (A.6), and
the corresponding scalars, q 7 and q$, are similar to p 7 and p c , respectively-
A.2 The Quadratic Spline Collocat ion Met hod
A.2.1 The Matrices
The matrices associated with the QSC method have narrower bandwidths than those of
their QSG counterparts. With biperiodic boundary conditions, the mass matrices in the
A- and O-directions are
- for i, j = 1; - - - , iV,\ and k, 1 = 1, - - - , No. Moreoever, P ' + ~ - PA*')^ and Q'+A =
( Q " ~ ' ) ~ . The first derivative matrices are defined by:
PA,., = B ~ P I ( T \ ~ ) , Q s t , , = P P ' ~ - & )
* p , , Q B ' ~ - A s
- I
for i, j = 1, - - - , N A and k,l = 1, - - ,Ne, where As = AX for the P-matrices and A6
for the Q-matrices. A h : P,~+A = , and Q++* = -(QP+')T- Finally, the
stiffness matrices are
A.2.2 Spectral Properties
The spectral properties of the above matrices are summarized in this section. As in the
QSG case, the vectors d k and Bk are eigenvectors of P, P.\ and Px*. The associated
eigenvalues, denoted by p k , p,\, and p.,,, , corresponding to both 9 k and &, are [6]
The matrices P*'~, p6+*, and P.?-'" satisfy (A.6) with p z and p z defined
below:
p h = COS (y) 3 - 2 kAX
p.\, - z- 4~ sin (-)
APPENDIX -4. THE DISCRETE OPERATORS
Similar results can be obtained in the 0-dimension-
A.3 The One-Step Optimal Quadratic Spline Collo-
cation Method
A.3.1 The Matrices
Perturbed first derivative operators are defined in the one-step OQSC method. In the
A-direct ion,
and piA+" - - - ( P ~ A + ~ ) '. Perturbed operators in the &direction are defined similarly.
A.3.2 Spectral Properties
The perturbed operators satisfy equations sirnilar to (A.6) with pC and p4 defined as
follows:
'+ PX" = 2 sin(kAX) - 6 sin (y)) -
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