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

✍ Scribed by I.P. Gavrilyuk; M. Hermann; M.V. Kutniv; V.L. Makarov

• Book ID: 104000842
• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 313 KB
• Volume: 23
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2010.01.016

✦ Synopsis

Nonlinear boundary value problem Two-point difference scheme Exact difference scheme Truncated two-point difference scheme a b s t r a c t

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 twopoint 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 be 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.