Numerical Initial Value Problems in Ordinary Differential Equations William Gear

Numerical Initial Value Problems in Ordinary Differential Equations by C. William Gear

Publisher: Prentice-Hall /
Year: 1971 /
ISBN: 0136266061 /
Description: Hardcover, Dust Jacket, Good/Good condition /
Condition: Previous owner\'s name, edge wear to jacket, no underlining, no highlighting /
Pages: 253 pages

Professor Gear treats all classes of step-by-step methods suitable to automatic integration of general problems.

No previous knowledge of computer programming is required, and the book is organized so that complex mathematical theory can be omitted.

Among the contents are chapters dealing with Euler method; higher order one-step methods; systems of equations and equations of order greater than one; convergence, error bounds and error estimates for one-step methods; the choice of step size and order; extrapolation methods; an introduction to multivalue and multistep methods; general multistep methods, order and stability; multivalue methods; existence, convergence and error estimates for multivalue methods; special methods for special problems; and choosing a method.

AMONG THE FEATURES

• Broad coverage includes methods for Stiff equations, multistep methods, convergence and error theory of predictor corrector methods, and extrapolation techniques.

• Provides complete FORTRAN subroutines for Runge-Kutta, extrapolation and multistep methods — both for general and Stiff problems.

• Offers both theoretical and computational problems at the end of each chapter.

• Presents many numerical examples of error behavior, instability effects, and round-off effects.

• Compares different methods numerically.

• Derives algorithms for controlling the step size and order of the method.

THE AUTHOR

C. WILLIAM GEAR is Professor of Computer Science and Applied Mathematics at the University of Illinois at Urbana-Champaign. This is his second published work. His first, published in 1969, dealt with computers and programming techniques.

CONTENTS

1 Introduction

1.1 The Problem to be Solved 1
1.2 Numerical Approximation of the Solution 7
1.3 An Illustration—the Euler Method 10
1.3.1 Error estimates 13
1.3.2 Comparison of error estimates with actual errors 15
1.3.3 Stability 16
1.3.4 Round-off errors 18
1.3.5 The perturbation due to the numerical approximation 21
Problems 23

2 Higher Order One-Step Methods

2.1 The Taylor\'s Series Method 25
2.2 Richardson Extrapolation to h = 0 26
2.3 Second Order Runge-Kutta Methods 27
2.4 Explicit Runge-Kutta Methods 31
2.4.1 The classical Runge-Kutta method 35
2.4.2 The Ralston Runge-Kutta method- 36
2.4.3 Butcher\'s results on the attainable order of Runge-Kutta methods 36
2.5 Implicit Runge-Kutta Methods 37
2.5.1 Practical considerations of implicit Runge-Kutta methods 39
2.6 Convergence and Stability 40
2.6.1 Stability regions for explicit Runge-Kutta methods 40
2.6.2 Stability regions for implicit Runge-Kutta methods 42
Problems 43

3 Systems of Equations and Equations of Order Greater than One

3.1 Application of One-Step Techniques to Systems of Equations 46
3.2 Reduction of a Higher Order Equation to a System of First Order Equations 47
3.3 Direct Methods for Higher Order Equations 47
3.3.1 Taylor\'s series methods 48
3.3.2 Runge-Kutta methods 48
Problems 50

4 Convergence, Error Bounds, and Error Estimates for One-Step Methods

4.1 Vector and Matrix Norms 53
4.2 Existence and the Lipschitz Condition 54
4.3 Convergence and Stability 55
4.4 Error Bounds and Order of Convergence 59
4.5 Asymptotic Error Estimates 61
4.5.1 The perturbation due to the numerical approximation 64
4.6 General Application of Error Bound and Estimate Theorems 66
4.6.1 Taylor\'s series methods 67
4.6.2 Runge-Kutta methods 67
4.6.3 The need for continuous derivatives 68
4.7 Variable Step Size 68 Problems 70

5 The Choice of Step Size and Order

5.1 The Choice of Order 73
5.2 Choice of Step Size 76
5.3 The Practical Control of Errors 79
5.4 Estimation of the Local Truncation Error 81
5.4.1 Step doubling 81
5.4.2 The Runge-Kutta-Merson method 85
Problems 86

6 Extrapolation Methods

6.1 Polynomial Extrapolation 87
6.1.1 Example of polynomial extrapolation 89
6.1.2 Round-off effects 89
6.1.3 Stability 89
6.1.4 Higher order methods 92
6.2 Rational Function Extrapolation 93 Problems 100

7 Multivalue or Multistep Methods— Introduction

7.1 Multivalue Methods 103
7.2 Explicit Multistep Methods—the Adams-Bashforth Method 104
7.2.1 Generating functions for the coefficients 108
7.2.2 Two other techniques for deriving the Adams-Bashforth methods 109
7.2.3 Truncation error in the Adams-Bashforth methods 110
7.3 Implicit Multistep Methods—the Adams-Moulton Method 111
7.4 Predictor-Corrector Methods 114
Problems 115

8 General Multistep Methods, Order, and Stability

8.1 The Order of a Multistep Method 117
8.1.1 Determination of cc if g is given and vice versa 119
8.1.2 The principal root of a method 120
8.2 Milne\'s Method 121
8.2.1 Stability of Milne\'s method for y\' = Ay 122
8.3 Stability of General Multistep Methods 124
8.3.1 Absolute stability 126
8.4 The Class of Three-Step Methods of Order Four 132
Problems 134

9 Multivalue Methods

9.1 Behavior of the Error 137
9.1.1 Stability of predictor-corrector methods 138
9.2 Equivalent Methods 142
9.2.1 Factors affecting the choice of representation 143
9.2.2 Adams\' methods in the backward difference representation 147
9.2.3 The Nordsieck form of Adams\' method 148
9.2.4 Modified multistep methods 150
9.2.5 Higher order equations 151
9.3 Automatic Control of Step Size and Order 155
Problems 167

10 Existence, Convergence, and Error Estimates for Multivalue Methods

10.1 Convergence and Stability 172
10.1.1 Stability 174
10.1.2 Order 180
10.1.3 Consistency and convergence 187
10.2 The Maximum Order of a Stable Multistep Method 194
10.3 Existence of Stable Multivalue Methods 198
10.4 Improved Order for Normal Form Multivalue Methods 200
10.5 Asymptotic Behavior of the Error 204 Problems 207

11 Special Methods for Special Problems

11.1 Stiff Equations 209
11.1.1 Multistep methods 212
11.1.2 A-stable methods 220
11.1.3 Methods based on a knowledge of df/dy 222
11.2 Algebraic and Singular Equations 223
11.3 Parameter Estimation 227
Problems 229

12 Choosing a Method

12.1 Effect of Future Developments 235

Bibliography
Index

nthdegree books