Numerical Methods for Ordinary Differential Equations: Initial Value Problems

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Springer Science & Business Media, Nov 11, 2010 - Mathematics - 271 pages

Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject.

It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples.

Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors.

The book covers key foundation topics:

o Taylor series methods

o Runge--Kutta methods

o Linear multistep methods

o Convergence

o Stability

and a range of modern themes:

o Adaptive stepsize selection

o Long term dynamics

o Modified equations

o Geometric integration

o Stochastic differential equations

The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com

 

Contents

1 ODEsAn Introduction
1
2 Eulers Method
19
3 The Taylor Series Method
33
Construction and Consistency
43
Convergence and ZeroStability
61
Absolute Stability
74
Systems of ODEs
95
Solving Implicit Methods
109
13 Modified Equations
177
14 Geometric Integration Part IInvariants
194
15 Geometric Integration Part IIHamiltonian Dynamics
207
16 Stochastic Differential Equations
225
A Glossary and Notation
242
B Taylor Series
247
C Jacobians and Variational Equations
251
D ConstantCoefficient Difference Equations
254

Order Conditions
123
10 RungeKutta MethodsII Absolute Stability
135
11 Adaptive Step Size Selection
144
12 LongTerm Dynamics
165
Bibliography
261
Index
267
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