A novel two-dimensional convection-diffusion finite-difference scheme

In tills article, we develop a two-dimensional finite-difference scheme for solving the convection-diffusion equation. The numerical method involves u sing transformation on the prototype scalar transport equation and transfer ring it to a Helmholtz equation. We apply the alternating-direction implici t scheme of Polezhaev to solve for the Helmholtz equation. As the key to su ccess ill simulating the convection-diffusion equation, we exploit the solu tion pertaining to the Helmholtz Equation in the coal se of scheme developm ent, 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 fall assessment of the proposed method. The results provide us with useful insights into the nature of the scheme, It is standard practice to v alidate the cone by investigating test problems which ale amenable to exact solutions to tbe working equation. Results show exact agreement for the on e-dimensional test problem and good agreement with the analytic solutions f or two-dimensional problems.