Numerical Simulation of Stiff Systems of Ordinary Differential Equations Using a Block Extended Backward Differentiation Scheme with a Stability Control Parameter

Abstract

Stiff systems of ordinary differential equations (ODEs) arise frequently in several areas of applied mathematics, science, and engineering, and their numerical integration requires methods with strong stability characteristics. In this paper, a block extended backward differentiation formula (BEBDF) with a stability control parameter is developed for the numerical solution of stiff systems of ODEs. The proposed method is constructed within the framework of block multistep methods, allowing the simultaneous computation of solution values at several grid points. A free parameter $\rho$ is incorporated into the formulation of the method to regulate and improve its stability behavior. The presence of this parameter provides additional flexibility in controlling the stability properties of the scheme, which is particularly beneficial when dealing with stiff problems. The nonlinear system arising from the implementation of the proposed method is solved using Newton’s iteration technique to ensure efficient and reliable convergence. The computational algorithm for the method is implemented in the Dev-C++ compiler environment, where the block structure of the scheme facilitates efficient numerical computation. Furthermore, the fundamental properties of the method, including consistency, zero-stability, and convergence, are analyzed to establish its theoretical reliability. Numerical experiments performed on selected stiff test problems demonstrate that the proposed method produces accurate and stable results. The findings indicate that the inclusion of the free parameter $\rho$ significantly enhances the stability control and overall performance of the method in the numerical solution of stiff systems of ODEs.