ebook img

Strong stability preserving Runge-Kutta and multistep time discretizations PDF

189 Pages·2011·3.032 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Strong stability preserving Runge-Kutta and multistep time discretizations

STRONG STABILITY PRESERVING RUNGE–KUTTA AND MULTISTEP TIME DISCRETIZATIONS 7498 tp.indd 1 12/20/10 3:48 PM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 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 7498 tp.indd 2 12/20/10 3:48 PM 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore. EH - Strong Stability.pmd 1 1/31/2011, 5:02 PM December5,2010 8:16 WorldScientificBook-9inx6in SSPbook In Memory of David Gottlieb November 14, 1944 - December 6, 2008 December5,2010 8:16 WorldScientificBook-9inx6in SSPbook TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk December5,2010 8:16 WorldScientificBook-9inx6in SSPbook 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. vii December5,2010 8:16 WorldScientificBook-9inx6in SSPbook 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

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.