A Comparative Study of High Order Modified Second Derivative Simpson’s Related Block Methods for Stiff Systems

Abstract

A continuous formulation based on the Modified Second Derivative Simpson’s Block Methods (MSDSBM) with off-grid points are developed and adapted to cope with the integration of stiff systems of Ordinary Differential Equations (ODEs). The LU Decomposition techniques was employed, which yields continuous formulation to derive theStandard Simpson’s Block Method (SSBM), Second Derivative Simpson’s Block Method (SDSBM), and the MSDSBM.This is achieved by combining the Modified Second Derivative Simpson’s Method (MSDSM) with other additional methods (obtained from the same continuous formulation) and applying them as numerical integrators by assembling them into a single block matrix equation. The basic stability properties of the block methods was investigated and found to be zero-stable, consistent, and convergent, and from their regions of absolute stability, they possess regions suitable for the solution of all stiff ordinary differential equations. Further investigation showed that the newly constructed methods are all A-stable and of high order. The performance of the methods was demonstrated on some numerical examples to show accuracy and computational efficiency.