Asymptotic error expansions for numerical solutions of one-dimensional problems with singularities
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.
scientific article; zbMATH DE number 91857