Introduction to Numerical Methods for Time Dependent Differential Equations E-Book

Heinz-Otto Kreiss

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 182

Veröffentlichungsjahr: 2014

Beschreibung

Introduces both the fundamentals of time dependent differential equations and their numerical solutions

Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs).

Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the theory of scalar equations, finite difference approximations, and the Explicit Euler method. Next, a discussion on higher order approximations, implicit methods, multistep methods, Fourier interpolation, PDEs in one space dimension as well as their related systems is provided.

Introduction to Numerical Methods for Time Dependent Differential Equations features:

• A step-by-step discussion of the procedures needed to prove the stability of difference approximations
• Multiple exercises throughout with select answers, providing readers with a practical guide to understanding the approximations of differential equations
• A simplified approach in a one space dimension
• Analytical theory for difference approximations that is particularly useful to clarify procedures

Introduction to Numerical Methods for Time Dependent Differential Equations is an excellent textbook for upper-undergraduate courses in applied mathematics, engineering, and physics as well as a useful reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs or predict and investigate phenomena from many disciplines.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Acknowledgements

Part I: Ordinary Differential Equations and Their Approximations

Chapter 1: First-Order Scalar Equations

1.1 Constant coefficient linear equations

1.2 Variable coefficient linear equations

1.3 Perturbations and the concept of stability

1.4 Nonlinear equations: the possibility of blow-up

1.5 Principle of linearization

Chapter 2: Method of Euler

2.1 Explicit Euler method

2.2 Stability of the explicit Euler method

2.3 Accuracy and truncation error

2.4 Discrete Duhamel’s principle and global error

2.5 General one-step methods

2.6 How to test the correctness of a program

2.7 Extrapolation

Chapter 3: Higher-Order Methods

3.1 Second-order Taylor method

3.2 Improved Euler’s method

3.3 Accuracy of the solution computed

3.4 Runge-Kutta methods

3.5 Regions of stability

3.6 Accuracy and truncation error

3.7 Difference approximations for unstable problems

Chapter 4: Implicit Euler Method

4.1 Stiff equations

4.2 Implicit Euler method

4.3 Simple variable-step-size strategy

Chapter 5: Two-Step and Multistep Methods

5.1 Multistep methods

5.2 Leapfrog method

5.3 Adams methods

5.4 Stability of multistep methods

Chapter 6: Systems of Differential Equations

Part II: Partial Differential Equations and Their Approximations

Chapter 7: Fourier Series and Interpolation

7.1 Fourier expansion

7.2 L2-norm and scalar product

7.3 Fourier interpolation

Chapter 8: 1-Periodic Solutions of time Dependent Partial Differential Equations with Constant Coefficients

8.1 Examples of equations with simple wave solutions

8.2 Discussion of well posed problems for time dependent partial differential equations with constant coefficients and with 1-periodic boundary conditions

Chapter 9: Approximations of 1-Periodic Solutions of Partial Differential Equations

9.1 Approximations of space derivatives

9.2 Differentiation of Periodic Functions

9.3 Method of lines

9.4 Time Discretizations and Stability Analysis

Chapter 10: Linear Initial Boundary Value Problems

10.1 Well-Posed Initial Boundary Value Problems

10.2 Method of lines

Chapter 11: Nonlinear Problems

11.1 Initial value problems for ordinary differential equations

11.2 Existence theorems for nonlinear partial differential equations

11.3 Nonlinear example: Burgers’ equation

Appendix A: Auxiliary Material

A.1 Some useful Taylor series

A.2 “” notation

A.3 Solution expansion

Appendix B: Solutions to Exercises

REFERENCES

Index

Introduction to Numerical Methods for Time Dependent Differential Equations

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Kreiss, H. (Heinz-Otto)  Introduction to numerical methods for time dependent differential equations / Heinz-Otto Kreiss, Tráskö-Storö Institute of Mathematics, Stockholm, Sweden, Omar Eduardo Ortiz, Facultad de Matmática Astronmía y Física, Universidad Nacional de Córdoba, Córdoba, Argentina.   pages cm  Includes bibliographical references and index.  ISBN 978-1-118-83895-2 (cloth)  1. Diffferential equations, Partial—Numerical solutions. I. Ortiz, Omar Eduardo, 1965– II. Title.  QA374.K915 2014515′.353dc232013042036

To our families

PREFACE

This book is based on the class notes of a course that H. Kreiss taught in the Department of Mathematics at UCLA in the year 1998. The original notes were then used by many other people. In particular, O. Ortiz used those original notes in a course taught in Fa.M.A.F, Universidad Nacional de Córdoba, in 2007 and 2010. The positive feedback from students taking these courses encouraged us to write the book.

Our intention was always to write a short book, suitable for an introductory, self-contained course that places emphasis on the fundamentals of time dependent differential equations and their relation to the numerical solutions of these equations.

The book is divided into two parts. The first part, from Chapter 1 to Chapter 6, deals with ordinary differential equations (ODEs) and their approximations. Chapter 1 is a simple presentation of the fundamental ideas in the theory of scalar equations. Chapter 2 is the core of the first part of the book, where most of the important concepts on finite-difference approximations are introduced and explained for the most basic method of all, the explicit Euler method. The remaining chapters deal with higher-order approximations, implicit methods, multistep methods, and systems of ODEs.

Our intention in this book is to emphasize the principles on which the theory is based. This is, one first needs to understand clearly the theory of scalar ordinary differential equations with constant coefficients. Then the variable coefficient problems are approached by appealing to the principle of frozen coefficients, which allows one to split the variable coefficient problem into many constant coefficient problems. Nonlinear problems are treated via the principle of linearization, which turns a nonlinear problem into a linear variable coefficient problem, which then decomposes into constant coefficient problems via the principle of frozen coefficients. For systems of ordinary differential equations we require that we can diagonalize the system, and then we just need to understand scalar equations.

The second part of the book deals with partial differential equations in one space dimension and their approximations. The basics of Fourier series and interpolation are presented in Chapter 7. Chapters 8, 9 and 10 are devoted to the concepts of well-posedness and numerical approximations for both Cauchy problems and initial boundary value problems. We start the discussion by treating in detail three basic equations: the one-way wave equation (or advection equation), the heat equation, and the wave equation. In Chapter 11, the final chapter, we develop the idea of “when” and “why” nonlinear differential problems can be thought of as perturbation of a numerically computed solution, thus making the approximations meaningful.

We want to make clear that one first needs to understand the theory of differential equations, including estimates of the solution, after which one can prove the stability of the difference approximations by similar estimates. Therefore, the usual way is that one gets from the theory existence during a finite time, then one approximates the problem by difference approximations and computes the solution for as long as the approximation remains stable.

Exercises, with most of their solutions provided in an appendix, were included based on the conviction that solving exercises and computing are essential to the learning process of this subject.

All the software used in the preparation of the manuscript is open-source software run under GNU-Linux. The typeseting was done in LaTeX. The numerical computations for examples and exercises were written in C and compiled with gcc. The plots were generated using Gnuplot and Gimp.

HEINZ-O. KIRESS and OMAR E. ORTIZ

CórdobaApril 2013.

ACKNOWLEDGMENTS

We would like to thank our students and colleagues for encouraging us to write this book, and other people for various contributions and help: in particular, Jarred Tanner for providing several solutions to exercises in the original Kreiss notes, Barbro Kreiss for being of invaluable help, and Gunilla Kreiss for carefully reviewing the manuscript and making suggestions.

This book was written partially with the support of grants 05/B454 from SeCyT, Universidad Nacional de Córdoba, and the Partner Group grant of the Max Planck Institute for Gravitational Physics, Albert-Einstein-Institute (Germany).

H.-O.K. and O.E.O.

PART I

ORDINARY DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS

CHAPTER 1

FIRST-ORDER SCALAR EQUATIONS

In this chapter we study the basic properties of first-order scalar ordinary differential equations and their solutions. The first and larger part is devoted to linear equations and various of their basic properties, such as the principle of superposition, Duhamel’s principle, and the concept of stability. In the second part we study briefly nonlinear scalar equations, emphasizing the new behaviors that emerge, and introduce the very useful technique known as the principle of linearization. The scalar equations and their properties are crucial to an understanding of the behavior of more general differential equations.

1.1 Constant coefficient linear equations

Consider a complex function y of a real variable t. One of the simplest differential equations that y can obey is given by