Two-step second-derivative high-order methods with two off-step points for solution of stiff systems

Abstract

Two-Step Second-Derivative High-Order Methods with off-step points suitable for the approximate numerical integration of stiff systems of first order initial value problems in ordinary differential equations are developed. The second derivative terms in the methods provide more freedom to derive a set of methods which are highly stable, convergent, with large regions of absolute stability. Although the cost of calculating the second derivatives may be higher than the first derivatives, the advantage gained makes them suitable for solving stable systems with large Lipschitz constants. Another requirement demands that the second derivative of high-order methods are to be solved iteratively rather than directly in the usual way of the linear multistep methods, however, efficient methods for doing this is provided from the associated continuous scheme of each method, by the evaluation of the continuous scheme at some off-step points. The derived methods are illustrated by the applications to some test problems of stiff systems and the numerical results obtained confirm the accuracy and efficiency of these methods.