A NOVEL TWO-DIMENSIONAL CONVECTION-DIFFUSION FINITE-DIFFERENCE SCHEME

Abstract

In this article, we develop a two-dimensional finite-difference scheme for solving the convection-diffusion equation. The numerical method involves using transformation on the prototype scalar transport equation and transferring it to a Helmholtz equation. We apply the alternating-direction implicit scheme of Polezhaev to solve for the Helmholtz equation. As the key to success in simulating the convection-diffusion equation, we exploit the solution pertaining to the Helmholtz equation in the course of scheme development, thereby providing high-level accuracy to the prediction. Since this is a new method developed for solving the model equation, it is illuminating to conduct modified equation analysis on the discrete equation in order to make a full assessment of the proposed method. The results provide us with useful insights into the nature of the scheme. It is standard practice to validate the code by investigating test problems which are amenable to exact solutions to the working equation. Results show exact agreement for the one-dimensional test problem and good agreement with the analytic solutions for two-dimensional problems.