The efficiency of second derivative multistep methods for the numerical integration of stiff systems

Abstract

A substantial increase in efficiency is achieved by the numerical integration methods which take advantage of the second derivative terms of the differential equation to be solved. The second-derivative of high order accuracy methods are stable, convergent and hence suitable for the numerical integration of stiff systems of initial value problems in ordinary differential equations. The unique feature of the paper is the idea of using all the set of collocation points as additional interpolation points. This desirable feature of the proposed approach actually widens the applicability of the methods, to include many other types of numerical integration methods and has many advantages, including didactic advantages. Furthermore, in this formulation symmetry is retained naturally by the integration identities as equal areas under the various segments of the solution curves over the integration interval. In this way the problem of overlap of solution models usually associated with multistep finite difference methods is overcome. The applications of the second derivative multistep integration methods on a significant class of problems found in the literature produce accurate solutions with low computational cost. Comparison of the efficiency curves obtained seems to be in better agreement with the exact solutions.