Analysis of Discretization Methods for Ordinary Differential Equations: (23 Springer Tracts in Natural Philosophy)

Due to the fundamental role of differential equations in science and engineering it has long been a basic task of numerical analysts to generate numerical values of solutions to differential equations. Nearly all approaches to this task involve a "finitization" of the original differential equation problem, usually by a projection into a finite-dimensional space. By far the most popular of these finitization processes consists of a reduction to a difference equation problem for functions which take values only on a grid of argument points. Although some of these finite­ difference methods have been known for a long time, their wide applica­ bility and great efficiency came to light only with the spread of electronic computers. This in tum strongly stimulated research on the properties and practical use of finite-difference methods. While the theory or partial differential equations and their discrete analogues is a very hard subject, and progress is consequently slow, the initial value problem for a system of first order ordinary differential equations lends itself so naturally to discretization that hundreds of numerical analysts have felt inspired to invent an ever-increasing number of finite-difference methods for its solution. For about 15 years, there has hardly been an issue of a numerical journal without new results of this kind; but clearly the vast majority of these methods have just been variations of a few basic themes. In this situation, the classical text­ book by P.

Table of Contents:
1 General Discretization Methods.- 1.1. Basic Definitions.- 1.2 Results Concerning Stability.- 1.3 Asymptotic Expansions of the Discretization Errors.- 1.4 Applications of Asymptotic Expansions.- 1.5 Error Analysis.- 1.6 Practical Aspects.- 2 Forward Step Methods.- 2.1 Preliminaries.- 2.2 The Meaning of Consistency, Convergence, and Stability with Forward Step Methods.- 2.3 Strong Stability of f.s.m..- 3 Runge-Kutta Methods.- 3.1 RK-procedures.- 3.2 The Group of RK-schemes.- 3.3 RK-methods and Their Orders.- 3.4 Analysis of the Discretization Error.- 3.5 Strong Stability of RK-methods.- 4 Linear Multistep Methods.- 4.1 Linear k-step Schemes.- 4.2 Uniform Linear k-step Methods.- 4.3 Cyclic Linear k-step Methods.- 4.4 Asymptotic Expansions.- 4.5 Further Analysis of the Discretization Error.- 4.6 Strong Stability of Linear Multistep Methods.- 5 Multistage Multistep Methods.- 5.1 General Analysis.- 5.2 Predictor-corrector Methods.- 5.3 Predictor-corrector Methods with Off-step Points.- 5.4 Cyclic Forward Step Methods.- 5.5 Strong Stability.- 6 Other Discretization Methods for IVP 1.- 6.1 Discretization Methods with Derivatives of f.- 6.2 General Multi-value Methods.- 6.3 Extrapolation Methods.