Stiffly Stable Diagonally Implicit Block Backward Differentiation Formula with Adaptive Step Size Strategy for Stiff Ordinary Differential Equations

Abstract

Experimental and theoretical evidence has shown that adaptive step size methods such as the backward differentiation formula (BDF) are more robust over a wider range of step sizes compared to those used in fixed step methods. Acknowledging the computational efficiency and accuracy obtained with such strategies, an adaptive step size version of the block backward differentiation formula (BBDF) in a diagonally implicit structure is proposed for solving stiff ordinary differential equations (ODEs), particularly in addressing the challenges posed by the chemical reaction problem within the domains of applied and industrial mathematics. The diagonally implicit structure with a lower triangular matrix and constant diagonal inputs offers significant advantages in evaluating the Jacobian and the lower-upper decomposition. The stability properties investigated show that the new class is zero-stable, A₀-stable and almost A−stable. Comparative evaluations reveal the superior performance of the proposed method compared to the existing fully implicit BBDF and ode15s conducted in MATLAB software.