Time Dependent Algorithm using Modified Three-Point Superclass of Block Backward Differentiation Formula for Solving Stiff Ordinary Differential Equations

In this paper, we present a modified three-point superclass of Block Backward Differentiation Formula (BBDF) for the efficient numerical solution of stiff systems of ordinary differential equations (ODEs). The principal enhancement of this work is a structural modification of the classical BBDF that forms a new parameterized superclass of methods, leading to improved stability and reduced error constants compared with the standard three-point BBDFs. The proposed scheme is formulated as a fully implicit block method capable of simultaneously producing three solution approximations within each integration step. A detailed theoretical analysis is conducted to establish the order of accuracy, consistency, zero-stability, and convergence of the method. Stability analysis based on Dahlquist test equation confirms that the scheme is A-stable and suitable for a wide range of stiff ODEs. The nonlinear systems arising from the implicit formulation are efficiently solved using Newton iteration. Numerical experiments on benchmark stiff ODE problems, including systems with rapidly decaying transient components, demonstrate that the modified three-point superclass BBDF achieves higher accuracy and lower computational cost when compared with existing block implicit methods, such as the standard BBDF and diagonally implicit three-point block BDF schemes. Overall, the proposed method provides a robust and computationally efficient alternative for the numerical integration of stiff ODEs, with potential applications in chemical kinetics, control theory, and biological modeling.