A Compact High-Order Finite Difference Method for Unsteady Convection-Diffusion Equation

Abstract

In this paper, a fourth-order compact finite difference method is proposed to solve the unsteady convection-diffusion equation. We first transform the convection-diffusion equation to a reaction-diffusion equation, which is then solved by a compact high-order method. The new method is unconditionally stable and fourth-order accurate in both temporal and spatial dimensions. It requires only a three-point stencil for a one-dimensional problem and a five-point stencil for a two-dimensional problem. Two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.

Keywords:

• Convection-diffusion equation
• Finite difference method
• Compact Scheme
• Unconditional stability
• Padé approximation

Acknowledgments

The research of the author is supported by the Natural Sciences & Engineering Research Council of Canada (RT734491) and MITACS project (POTSI). The author would like to thank the anonymous referees for their suggestions and comments on the revision of this manuscript.