Advances in Difference Equations
Volume 2006 (2006), Article ID 12167, 29 pages

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

I.P. Gavrilyuk1 , M. Hermann2 , M.V. Kutniv3 and V.L. Makarov4

1Berufsakademie Thüringen, Staatliche Studienakademie, Am Wartenberg 2, Eisenach 99817, Germany
2Institute of Applied Mathematics, Friedrich Schiller University, Ernst-Abbe-Platz 1-4, Jena 07740, Germany
3Lviv Polytechnic National University, 12 St. Bandery Street, Lviv 79013, Ukraine
4Department of Numerical Analysis, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv-4 01601, Ukraine


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 𝒪(|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.