Two-point difference schemes of an arbitrary given order of accuracy for nonlinear BVPs

In this paper we consider difference schemes for two-point BVPs for systems of first order nonlinear ODEs. 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 m. Here, we demonstrate that the TDS can he reduced to the numerical solution of some IVPs defined on each segment [x(j-1), x(j)] of the grid by an arbitrary IVP-solver of the order m. Using the difference schemes of the orders of accuracy m and m + 1 we develop an a posteriori error estimator for the numerical solution of the order m. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. It is based on embedded Runge-Kutta methods. Some numerical results confirming the efficiency of the algorithm are given. (C) 2010 Elsevier Ltd. All rights reserved.