Analysis of Discretization Methods for Ordinary Differential Equations

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...

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Main Author: Stetter, Hans J.
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1973, 1973
Edition:1st ed. 1973
Series:Springer Tracts in Natural Philosophy
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
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245 0 0 |a Analysis of Discretization Methods for Ordinary Differential Equations  |h Elektronische Ressource  |c by Hans J. Stetter 
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300 |a XVI, 390 p  |b online resource 
505 0 |a 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 Meth 
653 |a Mathematical analysis 
653 |a Ordinary Differential Equations 
653 |a Analysis (Mathematics) 
653 |a Differential equations 
653 |a Analysis 
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520 |a 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.