**Preprint series:** 06-25, Reports on Numerical Mathematics

I. P. Gavriljuk, M. Hermann, M. V. Kutniv, V. L. Makarov

**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:** A scalar boundary value problem (BVP)

for a second order differential equation on the infinite interval [0,1) is considered.

Under some natural assumptions it is shown that on an arbitrary finite grid there

exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme

of which the solution coincides with the projection of the exact solution of the given

differential equation onto the underlying grid. A constructive method is proposed to

derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where

n is a freely selectable natural number and [·] denotes the entire part of the expression

in brackets. The n-TDS has the order of accuracy Żn = 2[(n+1)/2], i.e., the global error

is of the form O(|h|Żn), where |h| is the maximum step size. The n-TDS is represented

by a system of nonlinear algebraic equations for the approximate values of the exact

solution on the grid. Iterative methods for its numerical solution are discussed. The

theoretical and practical results are used to develop a new algorithm which has all the

advantages known from the modern IVP-solvers. Numerical examples are given which

illustrate the theorems presented in the paper and demonstrate the reliability of the

new algorithm.

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

**Upload:** 2006-12-22

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