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COMPUTATIONAL
MATHEMATICS
Models, Methods, and Analysis
with MATLAB and MPI
© 2004 by Chapman & Hall/CRC
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
COMPUTATIONAL
MATHEMATICS
ROBERT E. WHITE
Models, Methods, and Analysis
with MATLAB and MPI
© 2004 by Chapman & Hall/CRC
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
© 2004 by Chapman & Hall/CRC
No claim to original U.S. Government works
International Standard Book Number 1-58488-364-2
Library of Congress Card Number 2003055207
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
Library of Congress Cataloging-in-Publication Data
White, R. E. (Robert E.)
Computational mathematics : models, methods, and analysis with MATLAB and MPI /
Robert E. White.
p. cm.
Includes bibliographical references and index.
ISBN 1-58488-364-2 (alk. paper)
1. Numerical analysis. 2. MATLAB. 3. Computer interfaces. 4. Parallel programming
(Computer science) I. Title.
QA297.W495 2003
519.4—dc21 2003055207
© 2004 by Chapman & Hall/CRC
Visit the CRC Press Web site at www.crcpress.com
Computational Mathematics:
Models, Methods and Analysis
with
M
ATLAB and MPI
R. E. White
Department of Mathematics
North Carolina State University
Updated on August 3, 2003
To Be Published by CRC Press in 2003
© 2004 by Chapman & Hall/CRC
white@math.ncsu.edu
Contents
List of Figures ix
List of Tables xi
Preface xiii
Introduction xv
1 Discrete Time-Space Models 1
1.1 Newton Cooling Models . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Heat Di
usion in a Wire . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Di
usion in a Wire with Little Insulation . . . . . . . . . . . . . 17
1.4 Flow and Decay of a Pollutant in a Stream . . . . . . . . . . . . 25
1.5 Heat and Mass Transfer in Two Directions . . . . . . . . . . . . . 32
1.6 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Steady State Discrete Models 51
2.1 Steady State and Triangular Solves . . . . . . . . . . . . . . . . . 51
2.2 Heat Di
usion and Gauss Elimination . . . . . . . . . . . . . . . 59
2.3 Cooling Fin and Tridiagonal Matrices . . . . . . . . . . . . . . . 68
2.4 Schur Complement . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.5 Convergence to Steady State . . . . . . . . . . . . . . . . . . . . 86
2.6 Convergence to Continuous Model . . . . . . . . . . . . . . . . . 91
3 Poisson Equation Models 99
3.1 Steady State and Iterative Methods . . . . . . . . . . . . . . . . 99
3.2 Heat Transfer in 2D Fin and SOR . . . . . . . . . . . . . . . . . 107
3.3 Fluid Flow in a 2D Porous Medium . . . . . . . . . . . . . . . . . 116
3.4 Ideal Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5 Deformed Membrane and Steep est Descent . . . . . . . . . . . . 130
3.6 Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . 138
v
© 2004 by Chapman & Hall/CRC
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