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

Qianqian Chu1,2, Guanghui Jin1, Jihong Shen2, Yuanfeng Jin1   

  1. 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.

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.

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