Development of the Established Redlich-Kister Finite Difference Solution with MKSOR Iteration for Solving One Dimensional Diffusion Problems

Abstract

This paper contributes a new method known as the second-order Redlich-Kister Finite Difference (RKFD) solution to the partial differential equation, especially for one-dimensional (1D) diffusion problems. All derivative terms for the proposed method are needed for the discretization process of the second-order RKFD. Arranging the derivative terms will lead to the second-order RKFD approximation equations. Then, this approximation equation is applied to solve the system of the RKFD equation. As the large-scale and sparse coefficient matrix is obtained, it will be solved iteratively to regulate the high computational complexity by using the Gauss-Seidel (GS), Kaudd Successive Over Relaxation (KSOR) and Modified Kaudd Successive Over Relaxation (MKSOR) iterative methods. All of those iterative methods are developed according to the matrix structure of the system and are applied to three examples of the proposed problem. As a result, MKSOR iterative method showed significant improvement in terms of performance efficiency by contrast to GS and KSOR iterative methods. The performance efficiency is measured by the number of iterations, execution time and maximum norm.