Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation

ABSTRACT

In this article, two characteristic block-centred finite difference schemes are introduced and analysed to solve the nonlinear convection-dominated diffusion equation. One scheme is a linear difference approximation which shows that the discrete $L^{\mathrm{\infty }}\left(L^2\right)$ and $L^2\left(H^1\right)$ errors are $O(\mathrm{\Delta }t+h^2)$, while the other is a two-grid scheme which demonstrates that the discrete $L^{\mathrm{\infty }}\left(L^2\right)$ and $L^2\left(H^1\right)$ errors are $O(\mathrm{\Delta }t+h^2+H^3)$, where h corresponds to a finer grid and H corresponds to a coarser grid. Error estimates with both schemes are established on a non-uniform rectangular grid. Finally, numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis and validate the efficiency of the two-grid method.

KEYWORDS:

• Characteristic block-centred finite difference
• nonlinear
• convection-dominated diffusion equation
• two-grid
• error estimates

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15
• 65M25
• 65M55

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China with No. 91330106.