Convergence and Order of the 2-Point Diagonally Implicit Block Backward Differentiation Formula with Two Off-Step Points

Abstract

The development and formulation of a most reliable and efficient numerical schemes for the integration of stiff systems of ordinary differential equations in terms of order, convergence, stability requirements, accuracy, and computational expense has been a major challenged in the study of modern numerical analysis. In this paper, the order and convergence properties of the 2-point diagonally implicit block backward differentiation formula with two off-step points for solving first order stiff initial value problems have been studied, the method was derived and found to be of order five. The necessary and sufficient conditions for the convergence of the method have also been established. It has shown that the 2-point diagonally implicit block backward differentiation formula with two off-step points is both consistent and zero stable, having satisfied these two conditions of consistency and that of zero stability, it is therefore concluded that the method converges and suitable for the numerical integration of stiff systems.