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Scientific Computation Editorial Board J.-J. Chattot, Davis, CA, USA P. Colella, Berkeley, CA, USA W. Eist, Princeton, NJ, USA R. Glowinski, Houston, TX, USA Y. Hussaini, Tallahassee, FL, USA P. Joly, Le Chesnay, France H.B. Keller, Pasadena, CA, USA J.E. Marsden, Pasadena, CA, USA D.I. Meiron, Pasadena, CA, USA O. Pironneau, Paris, France A. Quarteroni, Lausanne, Switzerland and Politecnico of Milan, Milan, Italy J. Rappaz, Lausanne, Switzerland R. Rosner, Chicago, IL, USA P. Sagaut, Paris, France J.H. Seinfeld, Pasadena, CA, USA A. Szepessy, Stockholm, Sweden M.F. Wheeler, Austin, TX, USA

For other titles published in this series, go to www.springer.com/series/718

David A. Kopriva

Implementing Spectral Methods for Partial Differential Equations Algorithms for Scientists and Engineers

Prof. Dr. David A. Kopriva Dept. of Mathematics Florida State University Tallahassee, FL 32306-4510 USA e-mail: [email protected]

ISBN 978-90-481-2260-8

e-ISBN 978-90-481-2261-5

DOI 10.1007/978-90-481-2261-5 Library of Congress Control Number: 2009922124 © Springer Science + Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To my Wife, my Mother, and in memory of my Dad.

Preface

This book is aimed to be both a textbook for graduate students and a starting point for applications scientists. It is designed to show how to implement spectral methods to approximate the solutions of partial differential equations. It presents a systematic development of the fundamental algorithms needed to write spectral methods codes to solve basic problems of mathematical physics, including steady potentials, transport, and wave propagation. As such, it is meant to supplement, not replace, more general monographs on spectral methods like the recently updated “Spectral Methods: Fundamentals in Single Domains” and “Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics” by Canuto, Hussaini, Quarteroni and Zang, which provide detailed surveys of the variety of methods, their performance and theory. I was motivated by comments that I have heard over the years that spectral methods are “too hard to implement.” I hope to dispel this view—or at least to remove the “too”. Although it is true that a spectral code is harder to hack together than a simple finite difference code (at least a low order finite difference method on a square domain), I show that only a few fundamental algorithms for interpolation, differentiation, FFT and quadrature—the subjects of basic numerical methods courses—form the building blocks of any spectral code, even for problems in complex geometries. I pres