A High Order Compact Difference Scheme for Solving the Unsteady Convection-Diffusion Equation

Abstract

In practical engineering applications, convection diffusion equations are generally used to describe the transport processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics, pollutant & sediment transport and chemical engineering. In this paper, a high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is firstly tested by a two-dimensional convection-diffusion equation with a Gaussian pulse type concentration. The L 2 error norms are used to measure differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the Taylor’s vortex problem is investigated by the proposed method and good agreement is obtained with the exact solutions. From the three test problems, it is shown that the proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems and also can be applied to many engineering problems.