Strong stability preserving Runge-Kutta and multistep time discretizations

189 Pages·2011·3.032 MB·English

by  Sigal Gottlieb, David I Ketcheson, Chi-Wang Shu

STRONG STABILITY
PRESERVING RUNGE–KUTTA
AND MULTISTEP TIME
DISCRETIZATIONS
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STRONG STABILITY
PRESERVING RUNGE–KUTTA
AND MULTISTEP TIME
DISCRETIZATIONS
Sigal Gottlieb
University of Massachusetts Dartmouth, USA
David Ketcheson
KAUST, Kingdom of Saudi Arabia
Chi-Wang Shu
Brown University, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Gottlieb, Sigal.
Strong stability preserving Runge-Kutta and multistep time discretizations / by Sigal Gottlieb,
David Ketcheson & Chi-Wang Shu.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-9814289269 (hard cover : alk. paper)
ISBN-10: 9814289264 (hard cover : alk. paper)
1. Runge-Kutta formulas. 2. Differential equations--Numerical solutions. 3. Stability.
I. Ketcheson, David I. II. Shu, Chi-Wang. III. Title.
QA297.5.G68 2011
518'.6--dc22
2010048026
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Printed in Singapore.
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In Memory of
David Gottlieb
November 14, 1944 - December 6, 2008
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Preface
Strong stability preserving (SSP) high order time discretizations were de-
veloped to ensure nonlinear stability properties necessary in the numerical
solutionofhyperbolicpartialdifferentialequationswithdiscontinuoussolu-
tions. SSP methods preservethe strong stability properties – in any norm,
seminorm or convex functional – of the spatial discretization coupled with
first order Euler time stepping. Explicit strong stability preserving (SSP)
Runge–Kutta methods have been employed with a wide range of spatial
discretizations, including discontinuous Galerkin methods, level set meth-
ods, ENO methods, WENO methods, spectral finite volume methods, and
spectral difference methods. SSP methods have proven useful in a wide
variety of application areas, including (but not limited to): compressible
flow,incompressibleflow,viscousflow,two-phaseflow,relativisticflow,cos-
mologicalhydrodynamics,magnetohydrodynamics,radiationhydrodynam-
ics, two-speciesplasmaflow, atmospherictransport,large-eddysimulation,
Maxwell’sequations,semiconductordevices,lithotripsy,geometricaloptics,
and Schr¨odinger equations. These methods have now become mainstream,
and a book on the subject is timely and relevant. In this book, we present
SSP time discretizations from both the theoretical and practical points of
view. Thoselookingforanintroductionto the subjectwillfindit inChap-
ters 1, 2, 8, 10, and 11. Those wishing to study the development and
analysis of SSP methods will find Chapters 3, 4, 5, 8, and 9 of particular
interest. Finally, those looking for practical methods to use will find them
in Chapters 2, 4, 6, 7, and 9.
WeareverygratefultoourcolleagueColinMacdonaldwhocontributed
tomuchoftheresearchinthisbook,especiallyinChapters5,7,and9. We
also wish to thank Randy LeVeque for his guidance and for encouraging
the collaboration that led to this book.
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viii Strong StabilityPreserving Time Discretizations
Wordsarenotsufficienttoexpressourthankstoourfamilies,whohave
shared in the process of writing this book. Their support and encourage-
ment made this work possible, and the joy they bring to our lives make it
more meaningful.
Much of the research that led to this book was funded by the U.S.
Air Force Office ofScientific Research,under grantsFA9550-06-1-0255and
FA9550-09-1-0208. We wish to express our gratitude to Dr. Fariba Fahroo
oftheAFOSRforhersupportofrecentresearchinthefieldofSSPmethods,
and for her encouragement of and enthusiasm about this book project.
ThisworkisdedicatedtothememoryofDavidGottlieb,whosewisdom,
kindness,andintegritycontinuetobeaninspirationtothosewhoknewhim.
Sigal Gottlieb, David Ketcheson, & Chi-Wang Shu
October 2010
December9,2010 14:56 WorldScienti(cid:12)cBook-9inx6in StrongStability-toc
Contents
Preface vii
1. Overview: The Development of SSP Methods 1
2. Strong Stability PreservingExplicit Runge{Kutta
Methods 9
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 SSP methods as convex combinations of Euler’s method:
the Shu-Osher formulation . . . . . . . . . . . . . . . . . . 12
2.4 Some optimal SSP Runge{Kutta methods . . . . . . . . . 15
2.4.1 A second order method . . . . . . . . . . . . . . . 15
2.4.2 A third order method . . . . . . . . . . . . . . . . 16
2.4.3 A fourth order method . . . . . . . . . . . . . . . 20
3. The SSP Coe(cid:14)cient for Runge{Kutta Methods 25
3.1 The modi(cid:12)ed Shu-Osher form . . . . . . . . . . . . . . . . 27
3.1.1 Vector notation . . . . . . . . . . . . . . . . . . . 29
3.2 Unique representations . . . . . . . . . . . . . . . . . . . . 30
3.2.1 The Butcher form . . . . . . . . . . . . . . . . . . 31
3.2.2 Reducibility of Runge{Kutta methods . . . . . . . 32
3.3 The canonical Shu-Osher form . . . . . . . . . . . . . . . 33
3.3.1 Computing the SSP coe(cid:14)cient . . . . . . . . . . . 37
3.4 Formulating the optimization problem . . . . . . . . . . . 39
3.5 Necessity of the SSP time step restriction . . . . . . . . . 40
ix