Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java, Maple, and MATLAB.pdf

Ordinary and Partial Differential

Equation Routines in C, C ++ ,

Fortran, Java ® , Maple ® , and M ATLAB ®

Ordinary and Partial Differential

Equation Routines in C, C ++ ,

Fortran, Java ® , Maple ® , and M ATLAB ®

H.J. Lee and W.E. Schiesser

CHAPMAN & HALL/CRC

A CRC Press Company

Boca Raton London New York Washington, D.C.

Java is a registered trademark of Sun Microsystems, Inc.

Maple is a registered trademark of Waterloo Maple, Inc.

MATLAB is a registered trademark of The MathWorks, Inc. For product information, please contact:

The MathWorks, Inc.

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Library of Congress Cataloging-in-Publication Data

Lee, H. J. (Hyun Jin)

Ordinary and partial differential equation routines in C, C++, Fortran, Java, Maple, and

MATLAB / H.J. Lee and W.E. Schiesser.

p. cm.

Includes bibliographical references and index.

ISBN 1-58488-423-1 (alk. paper)

1. Differential equations—Data processing. 2. Differential equations, Partial—Data

processing. I. Schiesser, W. E. II. Title.

QA371.5.D37L44 2003

515

¢.

352

0285—dc22

2003055809

This book contains information obtained from authentic and highly regarded sources. Reprinted material

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Preface

Initial value ordinary differential equations (ODEs) and partial differential

equations (PDEs) are among the most widely used forms of mathematics in

science and engineering. However, insights from ODE/PDE-based models

are realized only when solutions to the equations are produced with accept-

able accuracy and with reasonable effort.

Most ODE/PDE models are complicated enough (e.g., sets of simultane-

ous nonlinear equations) to preclude analytical methods of solution; instead,

numerical methods must be used, which is the central topic of this book.

The calculation of a numerical solution usually requires that well-

established numerical integration algorithms are implemented in quality li-

brary routines. The library routines in turn can be coded (programmed) in a

variety of programming languages. Typically, for a scientist or engineer with

an ODE/PDE- based mathematical model, ﬁnding routines written in a famil-

iar language can be a demanding requirement, and perhaps even impossible

(if such routines do not exist).

The purpose of this book, therefore, is to provide a set of ODE/PDE in-

tegration routines written in six widely accepted and used languages. Our

intention is to facilitate ODE/PDE-based analysis by using the library rou-

tines to compute reliable numerical solutions to the ODE/PDE system of

interest.

However, the integration of ODE/PDEs is a large subject, and to keep this

discussion to reasonable length, we have limited the selection of algorithms

and the associated routines. Speciﬁcally, we concentrate on explicit (nonstiff)

Runge Kutta (RK) embedded pairs. Within this setting, we have provided

integrators that are both ﬁxed step and variable step; the latter accept a user-

speciﬁed error tolerance and attempt to compute a solution to this required

accuracy. The discussion of ODE integration includes truncation error moni-

toring and control, h and p reﬁnement, stability and stiffness, and explicit and

implicit algorithms. Extensions to stiff systems are also discussed and illus-

trated through an ODE application; however, a detailed presentation of stiff

(implicit) algorithms and associated software in six languages was judged

impractical for a book of reasonable length.

Further, we have illustrated the application of the ODE integration routines

to PDEs through the method of lines (MOL). Brieﬂy, the spatial (boundary

value) derivatives of the PDEs are approximated algebraically, typically by

ﬁnite differences (FDs); the resulting system of initial-value ODEs is then

solved numerically by one of the ODE routines.

Thus, we have attempted to provide the reader with a set of computational

tools for the convenient solution of ODE/PDE models when programming

in any of the six languages. The discussion is introductory with limited math-

ematical details. Rather, we rely on numerical results to illustrate some basic

mathematical properties, and we avoid detailed mathematical analysis (e.g.,

theorems and proofs), which may not really provide much assistance in the

actual calculation of numerical solutions to ODE/PDE problems.

Instead, we have attempted to provide useful computational tools in the

form of software. The use of the software is illustrated through a small number

of ODE/PDE applications; in each case, the complete code is ﬁrst presented,

and then its components are discussed in detail, with particular reference

to the concepts of integration, e.g., stability, error monitoring, and control.

Since the algorithms and the associated software have limitations (as do all

algorithms and software), we have tried to point out these limitations, and

make suggestions for additional methods that could be effective.

Also, we have intentionally avoided using features speciﬁc to a particular

language, e.g., sparse utilities, object-oriented programming. Rather, we have

emphasized the commonality of the programming in the six languages, and

thereby illustrate how scientiﬁc computation can be done in any of the lan-

guages. Of course, language-speciﬁc features can be added to the source code

that is provided.

We hope this format will allow the reader to understand the basic elements

of ODE/PDE integration, and then proceed expeditiously to a numerical solu-

tion of the ODE/PDE system of interest. The applications discussed in detail,

two in ODEs and two in PDEs, can be used as a starting point (i.e., as tem-

plates) for the development of a spectrum of new applications.

We welcome comments and questions about how we might be of assis-

tance (directed to wes1@lehigh.edu). Information for acquiring (gratis) all the

source code in this book is available from http://www.lehigh.edu/˜ wes1/

wes1.html. Additional information about the book and software is available

from the CRC Press Web site, http://www.crcpress.com.

Dr. Fred Chapman provided expert assistance with the Maple program-

ming. We note with sadness the passing of Jaeson Lee, father of H. J. Lee,

during the completion of H. J. Lee’s graduate studies at Lehigh University.

H. J. Lee

W. E. Schiesser

Bethlehem, PA

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