Application of accelerated overrelaxation method to the numerical solution of elliptic partial differential equations

Abstract

In this paper, the finite difference method is employed to discretize the partial differential equation (PDE) through replacement of the PDE by a difference equation to be satisfied by the values of the vector of unknowns x at a finite set of points in the domain of the independent variable. This discretisation ultimately results in an associated linear system of equations Ax=b, where A is an n-square matrix, b is an n×1 column vector and x is the vector of unknowns. A large body of iterative methods for solving such linear systems abound, and several of them have been studied in order to improve on their robustness, convergence and suitability for specialised systems. One of such methods is the Accelerated Overrelaxation (AOR). The AOR is a two-parameter generalization of the classical Jacobi, Gauss-Seidel and Successive Overrelaxation (SOR) methods for the iterative solution of the linear system Ax=b. Here, the basics of the AOR method is established, suitable values are assigned to the parameters involved, and the method is applied to solve some partial differential equations of elliptic type. Results of numerical experiments proved the effectiveness of the method.