High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

Abstract

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

Keywords:

• Advection-diffusion-reaction
• nonlinear
• finite difference method
• backward differencing formula
• multigrid

2010 AMS SUBJECT CLASSIFICATIONS:

• 35K55
• 35K61
• 65M06
• 65M22
• 65M55

Acknowledgments

The authors would thank the editors and anonymous reviewers for their constructive comments and suggestions that help improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

All data are obtained through numerical simulation experiments, which are real and reliable.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

Supports of the study are from the National Natural Science Foundation of China [grant number 12161067], Natural Science Foundation of Ningxia, China [grant number 2022AAC02023], National Youth Top-notch Talent Support Program of Ningxia, China, and the First Class Discipline Construction Project in Ningxia Universities: Mathematics.