Generalized three-point difference schemes of high-order accuracy for systems of second-order nonlinear ordinary differential equations

Abstract

For systems of second-order nonlinear ordinary differential equations with the Dirichlet boundary conditions, we develop generalized three-point difference schemes of high-order accuracy on a nonuniform grid. The construction of the suggested schemes requires solving four auxiliary Cauchy problems (two problems for systems of nonlinear ordinary differential equations and two problems for matrix linear ordinary differential equations) on the intervals [x j−1, x j ] (forward) and [x j , x j+1] (backward) at each grid point; this is done at each step by any single-step method of accuracy order $$ \bar m $$ = 2[(m+1)/2]. (Here m is a given positive integer, and [·] is the integer part of a number.) We prove that such three-point difference schemes have the accuracy order $$ \bar m $$ for the approximation to both the solution u of the boundary value problem and the flux K(x)d u/dx at the grid points.