Analysis of Convergence and Stability Properties of Diagonally Implicit 3-Point Block Backward Differentiation Formula for First Order Stiff Initial Value Problems

Study’s Excerpt/Novelty This paper introduces a novel diagonally implicit 3-point block backward differentiation formula (BDF) for efficiently solving first-order stiff initial value problems. The method, proven to have an order of accuracy of 5, meets the necessary and sufficient conditions for convergence, including consistency and zero stability. Comparative numerical results highlight its superior performance in terms of maximum error and CPU time, demonstrating its advantage over existing methods and offering a robust solution for integrating stiff initial value problems. Full Abstract This paper comprehensively analyses the diagonally implicit 3-point block backward differentiation formula (BDF) for solving first-order stiff initial value problems. We establish the necessary and sufficient conditions for convergence, including consistency and zero stability, and derive the method's order of accuracy, which is found to be 5. Stability analysis reveals that the method is almost A-stable, with an absolute stability region plotted. A C programming language code is developed using Newton's Iteration for numerical implementation and compiled in the Microsoft Dev C++ compiler environment. Comparative numerical results demonstrate the superior performance of the proposed method over the existing fully implicit 3-point block backward differentiation formula (3BBDF) in terms of maximum error and CPU time. Therefore, this method offers a new and efficient numerical solution for integrating stiff initial value problems.