Numerical Methods for Unsteady Convection-diffusion Problems Based on Combining Compact Difference Schemes with Runge-Kutta Methods

Abstract

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in space and time. The one-dimensional unsteady convection-diffusion equation was solved by combing a compact difference scheme with the Runge-Kutta method. The combined method has fourth-order accuracy in space and time. To check the accuracy of the combined method, numerical experiments were carried out and comparisons were performed with the Crank-Nicolson method. The analysis results indicated that the combined method is numerically stable at low wave numbers and small Courant-Friedrichs-Lewy numbers. The combined method has higher accuracy than the Crank-Nicolson method.