Singly Diagonally Implicit Multistep Block Method for Numerical Solution of Stiff Ordinary Differential Equations

Abstract

Stiff ordinary differential equations (ODEs) pose significant challenges in numerical computation due to their rapid variations. Conventional numerical methods often suffer from instability and inefficiency when addressing such equations, which compromises accuracy and hinders progress in mathematical modelling. This research aims to develop an improved numerical method, specifically a singly diagonally implicit multistep block method to effectively solve stiff ODEs. The formulation, which incorporates the block backward differentiation formula (BBDF) generates formulae with absolute stability properties and offers versatility through a free parameter within a specific interval. The diagonally implicit structure characterized by a lower triangular matrix, allows sequential solving with reduced computational costs associated with factorization and back substitution during Newton iteration. The methodology involves approximating solutions at two points simultaneously using a fixed-step approach and conducting comprehensive stability analysis, including zero stability, absolute stability and convergence. Compared to existing BDF methods and MATLAB’s stiff solver, the proposed method demonstrates reliability in solving both linear and nonlinear systems across various stiffness levels. This new method proves particularly effective for stiff ODE systems, showcasing its potential for broad scientific and engineering applications.