Volume 36, Issue 3
Revisit of Semi-Implicit Schemes for Phase-Field Equations

Anal. Theory Appl., 36 (2020), pp. 235-242.

Published online: 2020-09

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• Abstract

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with $general$ $potential$ function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481),  which studies the semi-implicit scheme for the Allen-Cahn equation with $polynomial$ $potentials$.

• Keywords

Semi-implicit, phased-field equation, energy dissipation, maximum principle.

65M06, 65M12

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@Article{ATA-36-235, author = {Tao Tang , }, title = {Revisit of Semi-Implicit Schemes for Phase-Field Equations}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {3}, pages = {235--242}, abstract = {

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with $general$ $potential$ function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481),  which studies the semi-implicit scheme for the Allen-Cahn equation with $polynomial$ $potentials$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-SU12}, url = {http://global-sci.org/intro/article_detail/ata/18284.html} }
TY - JOUR T1 - Revisit of Semi-Implicit Schemes for Phase-Field Equations AU - Tao Tang , JO - Analysis in Theory and Applications VL - 3 SP - 235 EP - 242 PY - 2020 DA - 2020/09 SN - 36 DO - http://doi.org/10.4208/ata.OA-SU12 UR - https://global-sci.org/intro/article_detail/ata/18284.html KW - Semi-implicit, phased-field equation, energy dissipation, maximum principle. AB -

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with $general$ $potential$ function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481),  which studies the semi-implicit scheme for the Allen-Cahn equation with $polynomial$ $potentials$.

Tao Tang. (2020). Revisit of Semi-Implicit Schemes for Phase-Field Equations. Analysis in Theory and Applications. 36 (3). 235-242. doi:10.4208/ata.OA-SU12
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