A modified-upwind with block-centred finite difference scheme based on the two-grid algorithm for convection-diffusion-reaction equations

ABSTRACT

A modified-upwind with block-centred finite difference scheme on the basis of the two-grid algorithm is presented for the convection-diffusion-reaction equations. This scheme can keep second-order accuracy in spatial mesh sizes for both state variables and fluxes in the convection–diffusion–reaction problem. Moreover, the two-grid algorithm is constructed in order to solve semilinear convection-dominated problems efficiently, in which the main idea is to settle an original semilinear equation on the coarse space, and next to settle a linearized equation on the fine space. The error estimate of the method proposed in this paper is given through theoretical analysis. It is indicated that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=O\left(h^{1/2}\right)$. Thus, solving such a large-scale nonlinear problem is as easy as linearized problems. Besides, there are some numerical experiments to corroborate in practice that the algorithm is effective and robust to solve convection–diffusion–reaction problems.

KEYWORDS:

• Convection–diffusion– reaction equations
• modified-upwind scheme
• block-centred finite difference method
• two-grid method
• numerical experiment

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N06
• 65N12
• 65N15

Acknowledgments

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 author(s).

Additional information

Funding

The work is supported by China Postdoctoral Science Foundation [grant numbers 2021T140576 and 2020M672505], Shandong Provincial Natural Science Foundation [grant number ZR2019MA049].