Designing an implicit Block Backward Differentiation Formula (BBDF) for stiff ordinary differential equations

First-order Ordinary Differential Equations (ODEs) are often characterized by stiffness, especially in models that describe complex real-world processes. This work presents a four-point, fixed-coefficient, diagonally implicit block backward differentiation formula (4BBDF) of second order, developed to address the numerical challenges associated with stiffness. The formulation incorporates a diagonal matrix into the Lagrange interpolation polynomial and is constructed using Maple to ensure accuracy and stability. Newton’s method is used to handle the nonlinear systems that arise. The proposed 4BBDF method is mathematically verified to be consistent, zero-stable, A-stable, and of second-order accuracy. Its implementation in C++ shows improved computational efficiency, reducing the number of steps required by approximately 50% when compared to existing methods. These results indicate that the proposed scheme is a reliable and effective tool for solving stiffness in ODE.