Volume 3, Issue 4
A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

Lizhen Chen & Chuanju Xu

East Asian J. Appl. Math., 3 (2013), pp. 333-351.

Published online: 2018-02

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  • Abstract

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

  • Keywords

Cahn-Hilliard, time splitting schemes, spectral methods, error analysis.

  • AMS Subject Headings

65M06, 65M12, 65N30, 65N35, 76A05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-3-333, author = {}, title = {A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {3}, number = {4}, pages = {333--351}, abstract = {

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.150713.181113a}, url = {http://global-sci.org/intro/article_detail/eajam/10861.html} }
TY - JOUR T1 - A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation JO - East Asian Journal on Applied Mathematics VL - 4 SP - 333 EP - 351 PY - 2018 DA - 2018/02 SN - 3 DO - http://doi.org/10.4208/eajam.150713.181113a UR - https://global-sci.org/intro/article_detail/eajam/10861.html KW - Cahn-Hilliard, time splitting schemes, spectral methods, error analysis. AB -

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

Lizhen Chen & Chuanju Xu. (1970). A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation. East Asian Journal on Applied Mathematics. 3 (4). 333-351. doi:10.4208/eajam.150713.181113a
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