Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers

Abstract

Difference schemes for two-point boundary value problems for systems of first-order nonlinear 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 a priori possesses an arbitrary given order of accuracy ᏻ(|h| m ) with respect to 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 an m-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.

Compact difference schemes of high-order accuracy

“…The results of numerical experiments [23] show that the proposed two-point difference schemes are more efficient than the shooting method in the solution of boundary-value problems with large Lipschitz constants, boundary-value problems for stiff systems of ordinary differential equations, and problems with small parameter. A new algorithm of numerical solution of the boundary-value problems (35) with the use of two-point difference schemes with given accuracy and automatic choice of points of a grid was developed in [22].…”

Difference schemes for nonlinear BVPs on the half-axis

“…Thus, the assumptions of Theorem 2.1 are satisfied and problem (10), (11), under the assumptions (12), possesses a unique solution in Ω([0, ∞), r) which can be determined by the fixed point iteration.…”

“…For convenience of the readers and with the aim to focus the attention on the main aspects of our new algorithm, nearly all proofs are omitted. These proofs are given in our technical report [12].…”

“…Contrary to shooting methods where -in general -the errors are studied by an a posteriori error analysis the technique of EDS allows one to construct not only efficient numerical algorithms but also to carry out a rigorous a priori error analysis as well as to prove a variety of important qualitative statements (existence, uniqueness and convergence results, a priori estimates, etc.). EDS and n-TDS for systems of first order ODEs on a finite interval were recently studied in [11]. These methods are based on Runge-Kutta solvers for the associated IVPs.…”

Finite Difference Method for a Second-order Ordinary Differential Equation with a Boundary Condition of the Third Kind

“…One of the methods is the iterative method that involves the iterative process. Iterative methods such as the finite difference method [10][11][12], the shooting method [13,14], the method of exact and truncated (of arbitrary order of accuracy defined by the user) difference schemes [15] and the integral equation method [13, 16,17] have been developed for obtaining approximate solutions to the boundary value problems. Recently a finite difference scheme for secondorder boundary value problems with a third boundary condition was reported in [18].…”