Numerical methods for fluid dynamics

Texts in Applied Mathematics 32

Editors

J.E. Marsden

L. Sirovich

S.S. Antman

Advisors

G. Iooss

P. Holmes

D. Barkley

M. Dellnitz

P. Newton

For other titles published in this series, go to

http://www.springer.com/series/1214

Dale R. Durran

Numerical Methods for Fluid

Dynamics

With Applications to Geophysics

Second Edition

ABC

Dale R. Durran

University of Washington

Department of Atmospheric Sciences

Box 341640

Seattle, WA 98195-1640

USA

[email protected]

Series Editors

J.E. Marsden

Control and Dynamical Systems, 107-81

California Institute of Technology

Pasadena, CA 91125

USA

[email protected]

S.S. Antman

Department of Mathematics

and

Institute for Physical Science

and Technology

University of Maryland

College Park, MD 20742-4015

USA

[email protected]

L. Sirovich

Division of Applied Mathematics

Brown University

Providence, RI 02912

USA

[email protected]

ISSN 0939-2475

ISBN 978-1-4419-6411-3 e-ISBN 978-1-4419-6412-0

DOI 10.1007/978-1-4419-6412-0

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010934663

Mathematics Subject Classification (2010): 65-01, 65L04, 65L05, 65L06, 65L12, 65L20, 65M06,

65M08, 65M12, 65T50, 76-01, 76M10, 76M12, 76M22, 76M29, 86-01, 86-08

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Cover illustration: Passive-tracer concentrations in a circular flow with deforming shear, plotted at three

different times. This simulation was conducted using moderately coarse numerical resolution. A similar

problem is considered in Section 5.9.5.

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Preface

This book is a major revision of Numerical Methods for Wave Equations in

Geophysical Fluid Dynamics; the new title of the second edition conveys its broader

scope. The second edition is designed to serve graduate students and researchers

studying geophysical fluids, while also providing a non-discipline-specific introduc￾tion to numerical methods for the solution of time-dependent differential equations.

Changes from the first edition include a new Chapter 2 on the numerical solution

of ordinary differential equations (ODEs), which covers classical ODE solvers as

well as more recent advances in the design of Runge–Kutta methods and schemes

for the solution of stiff equations. Chapter 2 also explores several characterizations

of numerical stability to help the reader distinguish between those conditions suffi￾cient to guarantee the convergence of numerical solutions to ODEs and the stronger

stability conditions that must be satisfied to compute reasonable solutions to time￾dependent partial differential equations with finite time steps. Chapter 3 (formerly

Chapter 2) has been reorganized and now covers finite-difference schemes for the

simulation of one-dimensional tracer transport due to advection, diffusion, or both.

Chapter 4, which is devoted to finite-difference approximations to more general par￾tial differential equations, now includes an improved discussion of skew-symmetric

operators. Chapter 5, “Conservation Laws and Finite-Volume Methods,” includes

new sections on essentially nonoscillatory and weighted essentially nonoscillatory

methods, the piecewise-parabolic method, and limiters that preserve smooth ex￾trema. A section on the discontinuous Galerkin method now concludes Chapter 6,

which continues to be rounded out with discussions of the spectral, pseudospectral,

and finite-element methods. Chapter 7, “Semi-Lagrangian Methods,” now includes

discussions of “cascade interpolation” and finite-volume integrations with large time

steps. More minor modifications and updates have been incorporated throughout the

remaining chapters.

The majority of the schemes presented in this text were introduced in either the

applied mathematics or the atmospheric science literature, but the focus is not on

the details of particular atmospheric models but on fundamental numerical methods

that have applications in a wide range of scientific and engineering disciplines.

vii

viii Preface

The prototype problems considered include tracer transport, chemically reacting

flow, shallow-water waves, and the evolution of internal waves in a continuously

stratified fluid.

A significant fraction of the literature on numerical methods for these problems

falls into one of two categories: those books and papers that emphasize theorems and

proofs, and those that emphasize numerical experimentation. Given the uncertainty

associated with the messy compromises actually required to construct numerical

approximations to real-world fluid-dynamics problems, it is difficult to emphasize

theorems and proofs without limiting the analysis to classical numerical schemes

whose practical application may be rather limited. On the other hand, if one relies

primarily on numerical experimentation, it is much harder to arrive at conclusions

that extend beyond a specific set of test cases. In an attempt to establish a clear

link between theory and practice, I have tried to follow a middle course between

the theorem-and-proof formalism and the reliance on numerical experimentation.

There are no formal proofs in this book, but the mathematical properties of each

method are derived in a style familiar to physical scientists. At the same time, nu￾merical examples are included that illustrate these theoretically derived properties

and facilitate the comparison of various methods.

A general course on numerical methods for time-dependent problems might

draw on portions of the material presented in Chapters 1–6, and I have used sec￾tions from these chapters in a graduate course entitled “Numerical Analysis of

Time-Dependent Problems” that is jointly offered by the Department of Applied

Mathematics and the Department of Atmospheric Sciences at the University of

Washington. The material in Chapters 7 and 9 is not specific to geophysics, and

appropriate portions of these chapters could also be used in courses in a wide

range of disciplines. Both theoretical and applied problems are provided at the end

of each chapter. Those problems requiring numerical computation are marked by

an asterisk.

The portions of the book that are most explicitly related to atmospheric science

are portions of Chapter 1, the treatment of spherical harmonics in Chapter 6, and

Chapter 8. The beginning of Chapter 1 discusses the relation between the equa￾tions governing geophysical flows and other types of partial differential equations.

Switching gears, Chapter 1 then concludes with a short overview of the strategies for

numerical approximation that are considered in detail throughout the remainder of

the book. Chapter 8 examines schemes for the approximation of slow-moving waves

in fluids that support physically insignificant fast waves. The emphasis in Chapter 8

is on atmospheric applications in which the slow wave is an internal gravity wave

and the fast waves are sound waves, or the slow wave is a Rossby wave and the fast

waves are both gravity waves and sound waves.

Many numerical methods for the simulation of internally stratified flow require

the repeated solution of elliptic equations for pressure or some closely related vari￾able. Owing to the limitations of my own expertise and to the availability of other

excellent references, I have not discussed the solution of elliptic partial differential

equations in any detail. A thumbnail sketch of some solution strategies is provided

in Section 8.1.3; the reader is referred to Chapter 5 of Ferziger and Peri´c (1997)

Preface ix

for an excellent overview of methods for the solution of elliptic equations arising in

computational fluid dynamics and to Chapters 3 and 4 of LeVeque (2007) for a very

accessible and somewhat more detailed discussion.

I have attempted to provide sufficient references to allow the reader to further

explore the theory and applications of many of the methods discussed in the text,

but the reference list is far from encyclopedic and certainly does not include every

worthy paper in the atmospheric science or applied mathematics literature. Refer￾ences to the relevant literature in other disciplines and in foreign language journals

are rather less complete.1

The first edition of this book could not have been written without the gener￾ous assistance of several colleagues. Christopher Bretherton, in particular, provided

many perceptive answers to my endless questions. J. Ray Bates, Byron Boville,

Michael Cullen, Marcus Grote, Robert Higdon, Randall LeVeque, Christoph Sch¨ar,

William Skamarock, Piotr Smolarkiewicz, and David Williamson all provided very

useful comments on individual chapters. Many students used earlier versions of

this manuscript in my courses in the Department of Atmospheric Sciences at the

University of Washington, and their feedback helped improve the clarity of the

manuscript. Two students to whom I am particularly indebted are Craig Epifanio

and Donald Slinn. I am also grateful to Jim Holton for encouraging me to write the

first edition.

Peter Blossey made many important contributions to the second edition, includ￾ing performing the computations for Figures 5.24–5.28 and 7.3. Comments by

Catherine Mavriplis and Ram Nair helped improve the new section on discontin￾uous Galerkin methods. Joel Thornton helped me better grasp the fundamentals of

atmospheric ozone chemistry. Additional invaluable input was provided from many

readers of the first edition who were kind enough to send their comments and help

identify typographical errors.

It is my pleasure to acknowledge the many years of support for my numerical

modeling efforts provided by the Physical and Dynamic Meteorology Program of

the National Science Foundation. Additional significant support for my research

on numerical methods for atmospheric models has been provided by the Office of

Naval Research. Part of the first edition was completed while I was on sabbatical at

the Laboratoire d’A´erologie of the Universit´e Paul Sabatier in Toulouse, France, and

I thank Daniel Guedalia and Evelyne Richard for helping make that year productive

and scientifically stimulating.

As errors in the text are identified, they will be posted on the Web at http://

www. atmos.washington.edu/numerical.methods, which can be accessed directly

or via Springer’s home page at http://www.springer-ny.com. I would be most

grateful to be advised of any typographical or other errors by electronic mail at

[email protected].

Seattle, Washington, USA Dale R. Durran

1 Those not familiar with the atmospheric science literature may be surprised by the number of

references to Monthly Weather Review, which, despite its title, has become the primary American

journal for the publication of papers on numerical methods in atmospheric science.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction ................................................... 1

1.1 Partial Differential Equations: Some Basics..................... 2

1.1.1 First-Order Hyperbolic Equations ...................... 4

1.1.2 Linear Second-Order Equations in Two

Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Wave Equations in Geophysical Fluid Dynamics ................ 11

1.2.1 Hyperbolic Equations................................. 12

1.2.2 Filtered Equations ................................... 20

1.3 Strategies for Numerical Approximation ....................... 26

1.3.1 Approximating Calculus with Algebra ................... 26

1.3.2 Marching Schemes .................................. 30

Problems ...................................................... 33

2 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1 Stability, Consistency, and Convergence ........................ 36

2.1.1 Truncation Error ..................................... 36

2.1.2 Convergence ........................................ 38

2.1.3 Stability ............................................ 40

2.2 Additional Measures of Stability and Accuracy .................. 40

2.2.1 A-Stability .......................................... 41

2.2.2 Phase-Speed Errors .................................. 42

2.2.3 Single-Stage, Single-Step Schemes ..................... 44

2.2.4 Looking Ahead to Partial Differential Equations .......... 47

2.2.5 L-Stability .......................................... 48

2.3 Runge–Kutta (Multistage) Methods ........................... 49

2.3.1 Explicit Two-Stage Schemes........................... 50

2.3.2 Explicit Three- and Four-Stage Schemes................. 53

2.3.3 Strong-Stability-Preserving Methods .................... 55

2.3.4 Diagonally Implicit Runge–Kutta Methods............... 57

xi

xii Contents

2.4 Multistep Methods ......................................... 58

2.4.1 Explicit Two-Step Schemes............................ 58

2.4.2 Controlling the Leapfrog Computational Mode ........... 62

2.4.3 Classical Multistep Methods ........................... 67

2.5 Stiff Problems ............................................. 72

2.5.1 Backward Differentiation Formulae ..................... 73

2.5.2 Ozone Photochemistry ................................ 75

2.5.3 Computing Backward-Euler Solutions................... 77

2.5.4 Rosenbrock Runge–Kutta Methods ..................... 78

2.6 Summary ................................................. 81

Problems ...................................................... 85

3 Finite-Difference Approximations for One-Dimensional Transport. . . 89

3.1 Accuracy and Consistency ................................... 89

3.2 Stability and Convergence ................................... 92

3.2.1 The Energy Method .................................. 94

3.2.2 Von Neumann’s Method .............................. 96

3.2.3 The Courant–Friedrichs–Lewy Condition ................ 98

3.3 Space Differencing for Simulating Advection .................. 100

3.3.1 Differential–Difference Equations and Wave Dispersion ... 101

3.3.2 Dissipation, Dispersion, and the Modified Equation ....... 109

3.3.3 Artificial Dissipation ................................. 110

3.3.4 Compact Differencing ................................ 114

3.4 Fully Discrete Approximations to the Advection Equation ........ 117

3.4.1 The Discrete-Dispersion Relation ....................... 119

3.4.2 The Modified Equation ............................... 122

3.4.3 Stable Schemes of Optimal Accuracy ................... 123

3.4.4 The Lax–Wendroff Method ............................ 124

3.5 Diffusion, Sources, and Sinks ................................ 128

3.5.1 Advection and Diffusion .............................. 130

3.5.2 Advection with Sources and Sinks ...................... 137

3.6 Summary ................................................. 139

Problems ...................................................... 141

4 Beyond One-Dimensional Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.1 Systems of Equations ....................................... 147

4.1.1 Stability ............................................ 148

4.1.2 Staggered Meshes.................................... 153

4.2 Three or More Independent Variables .......................... 157

4.2.1 Scalar Advection in Two Dimensions ................... 157

4.2.2 Systems of Equations in Several Dimensions ............. 167

4.3 Splitting into Fractional Steps ................................ 169

4.3.1 Split Explicit Schemes ................................ 170

4.3.2 Split Implicit Schemes ................................ 173

4.3.3 Stability of Split Schemes ............................. 174

Contents xiii

4.4 Linear Equations with Variable Coefficients .................... 176

4.4.1 Aliasing Error ....................................... 178

4.4.2 Conservation and Stability............................. 184

4.5 Nonlinear Instability ........................................ 188

4.5.1 Burgers’s Equation ................................... 189

4.5.2 The Barotropic Vorticity Equation ...................... 193

Problems ...................................................... 197

5 Conservation Laws and Finite-Volume Methods . . . . . . . . . . . . . . . . . . . 203

5.1 Conservation Laws and Weak Solutions ........................ 205

5.1.1 The Riemann Problem ................................ 206

5.1.2 Entropy-Consistent Solutions .......................... 207

5.2 Finite-Volume Methods and Convergence ...................... 211

5.2.1 Monotone Schemes .................................. 213

5.2.2 Total Variation Diminishing Methods ................... 214

5.3 Discontinuities in Geophysical Fluid Dynamics ................. 217

5.4 Flux-Corrected Transport .................................... 221

5.4.1 Flux Correction: The Original Proposal .................. 222

5.4.2 The Zalesak Corrector ................................ 223

5.4.3 Iterative Flux Correction .............................. 226

5.5 Flux-Limiter Methods....................................... 226

5.5.1 Ensuring That the Scheme Is TVD ...................... 227

5.5.2 Possible Flux Limiters ................................ 230

5.5.3 Flow Velocities of Arbitrary Sign ....................... 234

5.6 Subcell Polynomial Reconstruction ........................... 235

5.6.1 Godunov’s Method ................................... 235

5.6.2 Piecewise-Linear Functions............................ 238

5.6.3 The Piecewise-Parabolic Method ....................... 240

5.7 Essentially Nonoscillatory and Weighted Essentially

Nonoscillatory Methods . . . .................................. 243

5.7.1 Accurate Approximation of the Flux Divergence .......... 244

5.7.2 ENO Methods ....................................... 246

5.7.3 WENO Methods ..................................... 249

5.8 Preserving Smooth Extrema .................................. 253

5.9 Two Spatial Dimensions ..................................... 255

5.9.1 FCT in Two Dimensions .............................. 256

5.9.2 Flux-Limiter Methods for Uniform Two-Dimensional Flow . 257

5.9.3 Nonuniform Nondivergent Flow ........................ 260

5.9.4 Operator Splitting .................................... 262

5.9.5 A Numerical Example ................................ 264

5.9.6 When Is a Limiter Necessary? ......................... 269

5.10 Schemes for Positive-Definite Advection ....................... 271

5.10.1 An FCT Approach ................................... 271

5.10.2 Antidiffusion via Upstream Differencing................. 273

5.11 Curvilinear Coordinates ..................................... 275

Problems ...................................................... 277

xiv Contents

6 Series-Expansion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

6.1 Strategies for Minimizing the Residual ......................... 281

6.2 The Spectral Method ........................................ 284

6.2.1 Comparison with Finite-Difference Methods ............. 285

6.2.2 Improving Efficiency Using the Transform Method ........ 293

6.2.3 Conservation and the Galerkin Approximation ............ 298

6.3 The Pseudospectral Method .................................. 299

6.4 Spherical Harmonics ........................................ 303

6.4.1 Truncating the Expansion ............................. 305

6.4.2 Elimination of the Pole Problem ........................ 308

6.4.3 Gaussian Quadrature and the Transform Method .......... 310

6.4.4 Nonlinear Shallow-Water Equations .................... 315

6.5 The Finite-Element Method .................................. 320

6.5.1 Galerkin Approximation with Chapeau Functions ......... 322

6.5.2 Petrov–Galerkin and Taylor–Galerkin Methods ........... 324

6.5.3 Quadratic Expansion Functions ........................ 327

6.5.4 Two-Dimensional Expansion Functions ................. 336

6.6 The Discontinuous Galerkin Method .......................... 339

6.6.1 Modal Implementation ................................ 341

6.6.2 Nodal Implementation ................................ 343

6.6.3 An Example: Advection .............................. 346

Problems ...................................................... 350

7 Semi-Lagrangian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

7.1 The Scalar Advection Equation ............................... 358

7.1.1 Constant Velocity .................................... 359

7.1.2 Variable Velocity .................................... 365

7.2 Finite-Volume Integrations with Large Time Steps .............. 369

7.3 Forcing in the Lagrangian Frame .............................. 372

7.4 Systems of Equations ....................................... 377

7.4.1 Comparison with the Method of Characteristics ........... 377

7.4.2 Semi-implicit Semi-Lagrangian Schemes ................ 379

7.5 Alternative Trajectories ..................................... 383

7.5.1 A Noninterpolating Leapfrog Scheme ................... 384

7.5.2 Interpolation via Parameterized Advection ............... 386

7.6 Eulerian or Semi-Lagrangian? ................................ 388

Problems ...................................................... 390

8 Physically Insignificant Fast Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

8.1 The Projection Method ...................................... 394

8.1.1 Forward-in-Time Implementation....................... 395

8.1.2 Leapfrog Implementation ............................. 397

8.1.3 Solving the Poisson Equation for Pressure ............... 398

8.2 The Semi-implicit Method ................................... 400

8.2.1 Large Time Steps and Poor Accuracy ................... 401

8.2.2 A Prototype Problem ................................. 403