A finite difference scheme with arbitrary mesh systems for solving high-order partial differential equations

Abstract

A finite difference technique for arbitrary irregular mesh systems is developed for numerical solutions of high-order boundary value problems. A successive differencing scheme is used to formulate the higher-order differencing equations. The details of the computational procedure for plate bending problems which are governed by a biharmonic equation are presented. In order to demonstrate the validity of the formulation procedure, numerical results obtained from the present technique were first compared with the analytical solutions for problems having regular and curved boundaries. For the regular boundary problem, the present results were also compared with that from the conventional finite difference method which is only suitable for treating regular meshes. Computations were then conducted to evaluate the accuracy of the solutions with different mesh arrangements. These studies have shown that, if well implemented, this method not only maintains the inherent simplicity of the finite difference method, but also puts it into a more programmable and flexible formulation so that the geometric generality, such as that in the finite element method, can be achieved.