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 mm. Here, we demonstrate that the TDS can be reduced to the numerical solution of some IVPs defined on each segment [xj−1,xj][xj−1,xj] of the grid by an arbitrary IVP-solver of the order mm. Using the difference schemes of the orders of accuracy mm and m+1m+1 we develop an a posteriori error estimator for the numerical solution of the order mm. 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.