A new unconditionally stable explicit scheme for the convection–diffusion equation with Robin boundary conditions

Abstract

An alternating direction explicit (ADE) scheme to solve the unsteady convection–diffusion equation with Robin boundary conditions is presented and discussed in this paper. It was derived based on the local series expansion method and proved unconditionally stable by von Neumann stability analysis. Thereafter, the ADE scheme is compared with the conventional schemes, and a comparison between the amplification factor of all schemes and the exact one shows that the proposed scheme can simulate well both convection- and diffusion-dominated problems. Finally, the proposed method was validated by a numerical experiment which indicates that, for large cell Reynolds numbers, the proposed scheme, which has unconditional stability, is more accurate than implicit schemes and most explicit schemes. It is also shown that the proposed scheme is simple to implement, economical to use, effective for dealing with Robin boundary conditions and easy to apply to multidimensional problems.

Keywords:

• convection–diffusion equation
• local series expansion method
• unconditionally stable
• alternating direction explicit scheme

2000 AMS Subject Classification :

• 35C10
• 41A58

CCS Category :

• G.1.8

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 50479053. The authors greatly appreciate Dr. Mark Walkley, University of Leeds, for his helpful suggestion and discussion. Also, the valuable comments from the anonymous reviewers are greatly acknowledged.