A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Abstract

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration  to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.