Volume 3, Issue 2
On Symmetry Breaking of Allen-Cahn

Dong Li, Chaoyu Quan, Tao Tang & Wen Yang

CSIAM Trans. Appl. Math., 3 (2022), pp. 221-243.

Published online: 2022-05

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  • Abstract

This paper is concerned with numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.

  • Keywords

Allen-Cahn equation, symmetry breaking, steady state.

  • AMS Subject Headings

35K55, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-3-221, author = {Dong and Li and and 23325 and and Dong Li and Chaoyu and Quan and and 23326 and and Chaoyu Quan and Tao and Tang and and 23327 and and Tao Tang and Wen and Yang and and 23328 and and Wen Yang}, title = {On Symmetry Breaking of Allen-Cahn}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2022}, volume = {3}, number = {2}, pages = {221--243}, abstract = {

This paper is concerned with numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0030}, url = {http://global-sci.org/intro/article_detail/csiam-am/20536.html} }
TY - JOUR T1 - On Symmetry Breaking of Allen-Cahn AU - Li , Dong AU - Quan , Chaoyu AU - Tang , Tao AU - Yang , Wen JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 221 EP - 243 PY - 2022 DA - 2022/05 SN - 3 DO - http://doi.org/10.4208/csiam-am.SO-2021-0030 UR - https://global-sci.org/intro/article_detail/csiam-am/20536.html KW - Allen-Cahn equation, symmetry breaking, steady state. AB -

This paper is concerned with numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.

Dong Li, Chaoyu Quan, Tao Tang & Wen Yang. (2022). On Symmetry Breaking of Allen-Cahn. CSIAM Transactions on Applied Mathematics. 3 (2). 221-243. doi:10.4208/csiam-am.SO-2021-0030
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