Solving three-dimensional (3D) Laplace equations by successive over-relaxation method

摘要

Motivated by the assertion that all physical systems exist in three space dimensions, and that representation in one or two space dimensions entails a large degree of approximations. The main objective of this paper is to extend the successive over-relaxation (SOR) method which is one of the widely used numerical methods in solving the Laplace equation, the most often encountered of the Elliptic partial differential equations (PDEs) in two dimensions to solving it in three dimensions. This is done by providing an easier procedure to obtain proper estimates to the SOR parameter and the stability criterion which are the two determinant elements used in facilitating convergence to the solution when solving PDEs by the SOR method. The hope is that, with the emergence of this finding, the representation of physical and environmental science problem will be closer to reality by representing them in three dimensions.