The Numerical Solution of Boundary Value Problems on 'Long' Intervals

摘要

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. The solution of these problems proceeds in two steps. The first is to cut the infinite interval at a finite, large enough point and to insert additional, so called asymptotic boundary conditions at the far (right) end; the second is to solve the resulting two point boundary value problem by a numerical method, for example a difference scheme. In this paper the Box-scheme is investigated. Numerical problems arise, because standard algorithms use too many grid points as the length of the interval increases. An 'asymptotic' a priori mesh size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed. Through investigating the conditioning of the (linearized) Box-scheme, we find that the solutions can be obtained safely by the Newton procedure when partial pivoting is employed.