An Accurate Finite-Difference Scheme on Second Order Partial Differential Equations in 3D

Abstract

A numerical technique is presented for the approximate solution of second‐order partial differential equations in three dimensions. With the temporary introduction of two unknown functions of the coordinates the initial equation is separated into three parts that are reduced to ordinary differential equations, one for each dimension, associated with a finite‐difference scheme. The use of suitable manipulations and the elimination of the unknown functions, gives finally a linear system of algebraic equations that is solved using an iterative method. The numerical technique is tested on an example of fluid mechanics: the equation of steady‐state molecular diffusion of a substance in a continuous medium moving in a straight tube of square cross‐section. The numerical results based on the present technique are more accurate than those of the standard relaxation treatment and 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.