Variable Order Difference Schemes for Nonlinear Two-Point BVPs

by    I. P. Gavrilyuk, M. Hermann, M. V. and Makarov Kutniv

Preprint series: 05-15, Reports on Numerical Mathematics

I. P. Gavrilyuk, M. Hermann, M. V. and Makarov Kutniv

MSC:
65L10 Boundary value problems
65L12 Finite difference methods
65L20 Stability and convergence of numerical methods
65L50 Mesh generation and refinement
34B15 Nonlinear boundary value problems
34L30 Nonlinear ordinary differential operators

Abstract: Two-point boundary value problems for a system of nonlinear first order ordinary differential equations are considered. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which possesses a prescribed order of accuracy O(|h|^m) w.r.t. the maximal step size |h|. This m-TDS represents a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. In the present paper new efficient methods for the implementation of a m-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.

Keywords: systems of nonlinear ordinary differential equations, nonlinear boundary-value problem, two-point difference scheme, exact difference scheme, truncated two-point difference scheme of an arbitrary given accuracy order, and fixed point iteration

Upload: 2005-11-10


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