Three-Point Difference Schemes of Variable Order for Nonlinear BVPs on the Half-Axis

Preprint series: 05-04, 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: The scalar BVP
$$\begin{array}{c} \bruch{{d^{2}u}}{{dx^{2}}} - m^{2}u = - f\left( {x,u} \right),\quad x \in \left( {0,\infty} \right), \\[0.2cm] u\left( {0} \right) = \mu_{1} ,\quad \quad \mathop {\lim}\limits_{x \to \infty} u\left( {x} \right) = 0, \end{array}$$
on the infinite interval $[0,\infty)$ is considered. Under some natural assumptions it is shown that on
an arbitrary \textit{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 onto the grid of the exact solution of the corresponding differential equation. A constructive algorithm is proposed to derive from the EDS a so-called truncated difference scheme (TDS) of a given rank $\bar{n}=2[(n+1)/2]$, provided that the right-hand side possesses $n$ continuous derivatives between a finite number of discontinuity points. Here $[\cdot]$ denotes the entire part of the expression in brackets. The $\bar{n}$-TDS possesses the order of accuracy $O(|h|^{\bar{n}})$ w.r.t. the maximal step size $|h|$. The $\bar{n}$-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. 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