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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 L. Sirovich Control and Dynamical Systems, 107-81 Division of Applied Mathematics California Institute of Technology Brown University Pasadena, CA 91125 Providence, RI 02912 USA USA [email protected] [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] 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 c Springer Science+Business Media, LLC 1999, 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 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. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To every hand that’s touched the Wall 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)foravery 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
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