A Continuous Class of A-Stable Block Generalized Backward Differentiation Formulae for Solving Stiff Problems of Ordinary Differential Equations

Abstract

This paper presents the development and analysis of a continuous class of A-stable Block Generalized Backward Differentiation Formulae (BBDF) for the numerical solution of stiff ordinary differential equations (ODEs). The proposed methods, denoted as 4SCBGBDF and 4SHBGBDF, extend the conventional BBDF framework by incorporating continuous interpolation functions, which facilitate the generation of dense output without incurring additional computational cost. The block structure of these methods enables the simultaneous computation of multiple solution points within a single step, thereby enhancing computational efficiency and solution accuracy. A rigorous stability analysis confirms the A-stability of both methods, affirming their suitability for stiff initial value problems. Numerical experiments conducted on standard benchmark stiff problems validate the theoretical properties and demonstrate the superior performance of the proposed methods in terms of stability, accuracy, and computational efficiency. The results underscore the potential of continuous A-stable BBDF schemes as robust and reliable tools for solving stiff systems arising in scientific and engineering applications.