An accurate numerical solver on second order PDEs with variable coefficients in three dimensions

Abstract

In the present paper a numerical technique is developed for the approximate solution of second-order partial differential equations (PDEs) with variable coefficients in three dimensions. With the temporary introduction of two unknown auxiliary functions of the coordinate system the initial equation is separated into three parts that are reduced to ordinary differential equations, one for each dimension, that are discretized with a finite difference scheme. The use of suitable manipulations and the elimination of the unknown auxiliary functions, gives finally a linear system of algebraic equations where the matrix of the coefficients of the unknowns is diagonally dominant, a prerequisite for the rapid convergence of the iterative procedure. The efficiency and accuracy of the proposed numerical scheme is validated by its application to two test problems of fluid mechanics which have exact solutions. The numerical results based on the present technique are more accurate than those obtained by either the standard relaxation treatment with central differences or the ADI method when the contribution of the first-derivative terms in the initial equation is dominant. In all cases the comparison of the numerical results with those of the analytical solution, demonstrates the reliability of the presented numerical code.