Asymptotic error expansions for numerical solutions of one-dimensional problems with singularities
type: Item
instance of: scholarly article
title: Asymptotic error expansions for numerical solutions of one-dimensional problems with singularities
zbMATH Open document ID: 0765.65068
DOI: 10.1016/0895-7177(91)90041-5
author: Wiemin Han
published in: Mathematical and Computer Modelling
publication date: 1993-01-16T00:00:00Z
review text: The Richardson extrapolation technique has been successfully applied to increase the accuracy of numerical solutions of a great variety of problems. This technique relies --- as is well-known --- on asymptotic expansions of the form (*) \(T(h)=T(0)+c_ 1 h^{\alpha_ 1}+c_ 2 h^{\alpha_ 2}+\dots\) where \(T(0)\) is the quantity to be computed, \(T(h)\) is its numerical approximation depending on a discretization parameter \(h\), \(c_ i\) are constants independent of \(h\) and \(0<\alpha_ 1<\alpha_ 2<\dots\;\). Applying this method to finite difference solutions of initial value problems of ordinary differential equations, \(T(0)\) represents the exact solution at some mesh point, \(T(h)\) is the finite difference approximation of the solution at this point, \(h\) is the meshsize. If the input data of an ordinary differential equation problem are smooth then the solution is smooth and an asymptotic expansion of the form (*) can be proved. The aim of the author's paper is to derive systematically asymptotic expansions of finite difference solutions for ordinary differential equation problems (initial value problems as well as two-point boundary value problems) with singular solutions. Numerical examples show a great improvement in the accuracy of numerical solutions by using the developed extrapolation technique.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 65B05
zbMATH DE Number: 91857
zbMATH Keywords: asymptotic error expansions
zbMATH Keywords: initial value problems
zbMATH Keywords: numerical examples
zbMATH Keywords: Richardson extrapolation
zbMATH Keywords: finite difference solutions
zbMATH Keywords: two-point boundary problems
zbMATH Keywords: singular solutions
reviewed by: Hans Ade
MaRDI profile type: Publication
full work available at URL: 0895-7177(91)90041-5
OpenAlex ID: W1977618270
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description: scientific article; zbMATH DE number 91857