A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

Abstract

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm.

KEYWORDS:

• Two-dimensional EFK equation
• block-centred finite difference method
• BDF2
• modified two-grid algorithm
• numerical experiment

AMS SUBJECT CLASSIFICATION (2010):

• 65N06
• 65N12
• 65N15

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped 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 Shandong Provincial Natural Science Foundation [grant number ZR2023MA052]; China Postdoctoral Science Foundation [grant numbers 2021T140576 and 2020M672505].