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Volume 39, Issue 5
Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation

Qianqian Chu, Guanghui Jin, Jihong Shen & Yuanfeng Jin

DOI: 10.4208/jcm.2002-m2019-0213

J. Comp. Math., 39 (2021), pp. 655-665.

Published online: 2021-08

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  • Abstract

We consider numerical methods to solve the Allen-Cahn equation using the second-order 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.

  • Keywords

Allen-Cahn Equation, Crank-Nicolson scheme, Maximum principle, Convergence.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

1214181195@qq.com (Qianqian Chu)

jinguanghui@ybu.edu.cn (Guanghui Jin)

shenjihong@hrbeu.edu.cn (Jihong Shen)

yfkim@ybu.edu.cn (Yuanfeng Jin)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-655, author = {Chu , QianqianJin , GuanghuiShen , Jihong and Jin , Yuanfeng}, title = {Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {5}, pages = {655--665}, abstract = {

We consider numerical methods to solve the Allen-Cahn equation using the second-order 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2002-m2019-0213}, url = {http://global-sci.org/intro/article_detail/jcm/19381.html} }
TY - JOUR T1 - Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation AU - Chu , Qianqian AU - Jin , Guanghui AU - Shen , Jihong AU - Jin , Yuanfeng JO - Journal of Computational Mathematics VL - 5 SP - 655 EP - 665 PY - 2021 DA - 2021/08 SN - 39 DO - http://doi.org/10.4208/jcm.2002-m2019-0213 UR - https://global-sci.org/intro/article_detail/jcm/19381.html KW - Allen-Cahn Equation, Crank-Nicolson scheme, Maximum principle, Convergence. AB -

We consider numerical methods to solve the Allen-Cahn equation using the second-order 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.

Qianqian Chu, Guanghui Jin, Jihong Shen & Yuanfeng Jin. (2021). Numerical Analysis of Crank-Nicolson Scheme for the Allen-Cahn Equation. Journal of Computational Mathematics. 39 (5). 655-665. doi:10.4208/jcm.2002-m2019-0213
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