High Order Multi-block Boundary-value Integration Methods for Stiff ODEs

Abstract

In this paper, we present a new family of multi-block boundary value integration methods based on the Enright second derivative type-methods for differential equations. We rigorously show that this class of multi-block methods are generally $A_{k_1,k_2}$-stable for all block number by verifying through employing the Wiener-Hopf factorization of a matrix polynomial to determine the root distribution of the stability polynomial. Further more, the correct implementation procedure is as well determine by Wiener-Hopf factorization. Some numerical results are presented and a comparison is made with some existing methods. The new methods which output multi-block of solutions of the ordinary differential equations on application, and are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output a block of solutions per step. The second derivative multi-block boundary value integration methods are a new approach at obtaining very large scale integration methods for the numerical solution of differential equations.