Multiple Off-Grid Hybrid Block Simpson's Methods for Solution of Stiff Ordinary Differential Equations

Althemai J. M., Skwame Y, Donald J. Z.

Abstract: This paper is concerned with the construction of two-step hybrid block Simpsons methods with two and three off-grid points for the solutions of stiff systems of ordinary differential equations (ODEs). This is achieved by transforming a k-step multi-step method into continuous form and evaluating at various grid points to obtain the discrete schemes. The discrete schemes are applied each as a block for simultaneous integration. Each block matrix equation is A-stable and of order [5, 5, 5, 6]T and [6, 6, 6, 6, 6]T respectively. These orders are achieved by the aid of Maple13 software program. The performance of the methods is demonstrated on some numerical experiments. The results revealed that the hybrid block Simpsons method with two-off grid points was more efficient than that hybrid block method with three off-grid points on mildly stiff problems.

Keywords: hybrid method, off-step point, blocks method, first order system, multi-step method