NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

doi:10.4208/jcm.2002-m2019-0213

Journal of Computational Mathematics ›› 2021, Vol. 39 ›› Issue (5): 655-665.DOI: 10.4208/jcm.2002-m2019-0213

NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

Qianqian Chu^1,2, Guanghui Jin¹, Jihong Shen², Yuanfeng Jin¹

1. Department of Mathematics, Yanbian University, Yanji 133002, China;
2. Department of Mathematics Science, Harbin Engineering University, Harbin 150001, China

Received:2019-09-18 Revised:2020-01-19 Online:2021-09-15 Published:2021-10-15
Supported by:
This work was supported by National Natural Science Foundation of China (No. 11761074), the projection of the Department of Science and Technology of Jilin Province for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project.

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Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin. NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION[J]. Journal of Computational Mathematics, 2021, 39(5): 655-665.

We consider numerical methods to solve the Allen-Cahn equation using the secondorder Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in L_∞ norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.

Key words: Allen-Cahn Equation, Crank-Nicolson scheme, Maximum principle, Convergence

CLC Number:

65M06
65M12

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NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

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