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Volume 14, Issue 3
Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

Yongho Choi, Yibao Li, Chaeyoung Lee, Hyundong Kim & Junseok Kim

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 797-810.

Published online: 2021-06

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  • Abstract

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.

  • AMS Subject Headings

65M06, 65Z05, 68U20

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-797, author = {Choi , YonghoLi , YibaoLee , ChaeyoungKim , Hyundong and Kim , Junseok}, title = {Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {3}, pages = {797--810}, abstract = {

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0155}, url = {http://global-sci.org/intro/article_detail/nmtma/19198.html} }
TY - JOUR T1 - Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces AU - Choi , Yongho AU - Li , Yibao AU - Lee , Chaeyoung AU - Kim , Hyundong AU - Kim , Junseok JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 797 EP - 810 PY - 2021 DA - 2021/06 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0155 UR - https://global-sci.org/intro/article_detail/nmtma/19198.html KW - Allen-Cahn equation, conservative Allen-Cahn equation, Laplace-Beltrami operator, triangular surface mesh, hybrid numerical method, PDE on surface. AB -

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.

Yongho Choi, Yibao Li, Chaeyoung Lee, Hyundong Kim & Junseok Kim. (2021). Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces. Numerical Mathematics: Theory, Methods and Applications. 14 (3). 797-810. doi:10.4208/nmtma.OA-2020-0155
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