A two-point difference scheme of arbitrary given accuracy order for BVPs for systems of first order nonlinear ODEs

by    V.L.Makarov, I.P.Gavryliuk, M.V.Kutniv, M.Hermann

Preprint series: 03-03 , Reports on Numerical Mathematics

MSC:
65L10 Boundary value problems
65L12 Finite difference methods
65L20 Stability and convergence of numerical methods
65L50 Mesh generation and refinement
65L70 Error bounds
34B15 Nonlinear boundary value problems

Abstract: We consider two-point boundary value problems for systems of first order nonlinear ordinary differential equations. Under natural conditions we show that on an arbitrary grid there exists a unique two-point exact difference scheme (EDS), i.e. a difference scheme which solution coincides with the projection onto the grid of the exact solution of the corresponding system of differential equations. A constructive algorithm is proposed in order to derive from EDS a so-called truncated difference scheme of an arbitrary given rang m (m-TDS) possessing the accuracy order O(|h|^m) w.r.t. the maximal step size |h|. The m-TDS represents a system of nonlinear algebraic equations w.r.t. the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Analytical and numerical examples are given which illustrate the theorems proved.

Keywords: systems of nonlinear ordinary differential equations, difference scheme, exact difference scheme, truncated difference scheme of an arbitrary given accuracy order,

Upload: 2003-03-03


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