HIGH-ORDER SPATIAL DISCRETIZ- TIO ON METHODS ......From an eigenvalue analysis of our simplified...

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

National Library 1*1 of Canada Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographic Services services bibliographiques

395 Wellington Street 395. rue Wellington Ottawa ON K1A O N 4 ûttawa ON K1A ON4 Canada Canada

The author has granted a non- exclusive licence allowing the National L h r q of Canada to reproduce, loaq distribute or sell copies of this thesis in microfom, paper or electronic formats.

The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts f h m it rnay be printed or otherwise reproduced without the author's permission.

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/nlm, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation,

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


Group veloci ties of gravity waves in the &direction for the two-step OQSC



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


Group velocities of gravity waves in the 8-direction for the tww-step OQSC



. . . . . . . . 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-


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.


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


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)


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 ) ~


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.


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


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


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.


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


(S. 14)

Figure 8.10: Group velocities of gravity waves in the A-direction for the two-step OQSC


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


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


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.


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:



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


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.


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


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.


The perturbed operators satisfy equations sirnilar to (A.6) with pC and p4 defined as

follows:

'+ PX" = 2 sin(kAX) - 6 sin (y)) -

Bibliography

[l] G. K. Batchelor. An Introduction to Fiuid Dynamics. Cambridge University Press,

1992.

[2] J. R. Bates and A. McDondd. &1 ultiply-upstream, semi-Lagrangian advection

schemes: analysis and application to a multi-level primitive equation model. Month./

W-eather Review, 110:1S:31--1S4'2, 19S2.

[3] G. J. Boer and B. Denis. Numerical convergence of the dynamics of a gcm. Workshop

on Test Cases for Dynamics Cores of Atmospheric General Circulation Models, June

1996.

[4] John P. Boyd. Equatorial solitary waves. Part 1: Rossby solitons. Journal of Physicai

Oceanography, 10:1699-1717, November 1980.

[ 5 ] John P. Boyd. The nonlinear equatorial Kelvin wave. Journal of Physicai Oceanog-

raphy, 10:l-11, January 1980.

[6] C. C. Christara. Quadratic spline collocation methods for ellip tic partial differential

equat ions- BIT, 34(1):33-61, 1994.

[7] Christina C. Christara and Barry Smith. Multigrid and multilevel methods for

quadratic spline collocation. BIT, 37(4):781-803, 1997.

[SI Anthanasia Constas. Fast Fourier transform solvers for quadratic spline collocation.

MSc. Thesis, Department of Computer Science, University of Toronto, Toronto,

Ontario, Canada, 1996.

[9] Jean Côté. Variable resolution techniques for weather prediction. Meteornlogy and

Atmospheric Physics, 6331-38, 1997.

[IO] Jean Côté, Jean-Guy Desmarais, Sylvie Gravel, André Méthot, Alain Patoine,

Michel Roch, and Andrew Staniforth. The operational CMC-MRB Global En-

vironment Multiscale (GEM) model. Part II: Results. Monthly Wenther Review,

126:139'7-1416, June 1998.

[ I l ] Jean Côté, Sylvie Gravel, André Méthot, Alan Patoine, Michel Roch, and Andrew

Staniforth. The operational CMC-MRB Global Environmental Multiscale (GEM)

model. Part 1: Design considerations and formulation. Monthly Weather Review,

1%: L3ï3-1393t June 1998.

[12] Jean Côté and Andrew Stanifort h. A two-time-level semi-Lagrangian semi-implicit

scheme for spectral methods- iblonthly CVeather Reuiew, 116:2003-2012, August 1988.

[13] Jean Côté and Andrew Staniforth. An accurate and efficient finite-element global

model of the shallow water equations. Monthly Weather Reuiew, 1 lS:L'ïO?-i)'il'7,

1990.

[14] Car1 de Boor. A Practical Guide to Splines. New York: Springer-Verlag, 197s.

[L5] Dale R. Durran. Numerical iVfethods for Waue Equations in Geophysical Flîlid Dy-

namics. New York: Springer, 1999.

[16] Francis X. Giraldo and Beny Neta. Stability analysis for Eulerian and semi-

Lagrangian finite element rnethods of the advection-diffusion equation. Cornputers

and Mathematics with Applications, 38(2):97-112, 1997.

George J. Haltiner and Roger T. Williams. Numerical Prediction and Dynamic

Meteorology. John Wiley and Sons, 19SO.

Isaac Held and Max J, Suarez. A proposa1 for the intercomparison of the dynamical

cores of atmospheric general circulation models. Bulletin of American &le teorological

Society, pages 1825-1830, 1994.

James R, Holton. An Introduction to Dynamic Meteorology. San Diego Academic

Press, 1992.

E. N. Houstis, C. C. Christara, and J. R. Rice. Quadratic-spline collocation meth-

ods for two-point boundary value problems. International Journal for Numerical

Methods in Engineering, 26:935-953, 1988.

Mohamed Iskandarani, Dale B. Haidvogel, and John P. Boyd. A staggered spectral

elernent model wit h application to the ocean shallow water equations. International

Journal for Nwmerical Methods in Fluids, 10:393-414, 1995.

Hong Ma. A spectral elernent basin model for the shallow water equations. Journal

of Computational Physics, 109: 133-149, 1992.

M. J. Marsden. Quadratic spline interpolation. Bulletin of American Mathematics

Society, 30:903-906, 1974.

A. McDonald. Accuracy of multiply-upstream, semi-Lagrangian advective schemes.

Monthly Weather Review, 112: 1267-1275, 1954.

K. S. Ng. Quadratic spline collocation methods for systems of elliptic PDEs. M.Sc.

Thesis, Depart ment of Computer Science, University of Toronto, Toronto, Ontario,

Canada, 67 pgs., April 2000.

Harold Ritchie. Semi-Lagrangian advection on a Gaussian grid. Monthly Weather

Reuiew, ll5:6OS-619, 1957.

[27] André Robert. A stable numerical integrat ion scheme for the primitive meteorolog-

ical equations. Atmosphere-Ocean, 19(1):3546, l 9 S l .

[2S] Daniel T. Le Roux, Charles A. Lin, ând Andrew S taniforth. A semi-impiicit semi-

Lagrangian hiteelement shallow-water ocean model. Mon thZy Weather Review,

126:1931-1951, July 1998-

[-91 R. D. Russell and L. F. Shampine. A collocation method for boundary vdue prob-

lems. Numen-sche Mathematik, 19:l-2S1 1972,

[30] Andrew S taniforth and Herschel Mitchell. A semi-implicit finite-element barotropic

model. Monthly Weather Review, pages 154-169, February 1977.

[3 11 Andreiv S t anifort h and Clive Temperton. Semi-implicit semi-lagrangian integrat ion

schemes for barotropic finite-element regional model. Monthly Weather Review,

111:207S-2090, Novernber 19S6.

[31] Paul N. Swarzt rauber. Spectral transform met hods for solving the shallow-water

equations on the sphere. Monthly W e a ther Reuiew, 124:730-743, April 1996.

[3:3] Monique Tanguay, André Robert, and René Laprise. A semi-impiicit semi-

Lagrangian fdly compressible regional forecast model. Monthly Weather Reuiew,

11S:1970-19SO, October 1990.

[34] S . Thomas, J . Côté, and -4. Staniforth. A semi-implicit semi-Lagrangian shallow-

cvater model for massively parallel processors. Manuscript, 1993.

[35] D. L. Williamson and R. Laprise. Numerical approximations for global atmospheric

general circuiat ion models. Num erical Modelling of the Global A tmosphere in the

Climate System, 1000.

[36] David L. Williamson, John B. Drake, James J. Hack, Rudiger Jakob, and Paul N.

Swarztrauber. A standard test set for numerical approximations to the shallow water

equations in spherical geometry. Journal of Cornpututional Physics, 102(1):211 - 224,

1992.

Date post:	25-Dec-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

HIGH-ORDER SPATIAL DISCRETIZ- TIO ON METHODS ......From an eigenvalue analysis of our simplified...

Documents