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Volume 15, Issue 1-2
Uniform $L^p$-Bound of the Allen-Cahn Equation and Its Numerical Discretization

Jiang Yang, Qiang Du & Wei Zhang

Int. J. Numer. Anal. Mod., 15 (2018), pp. 213-227.

Published online: 2018-01

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We study uniform bounds associated with the Allen-Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, shares the uniform $L^p$-bound for all even $p$, which also leads to the maximum principle. In comparison, a couple of other spatial discretization schemes, namely the Fourier spectral Galerkin method and spectral collocation method preserve the $L^p$-bound only for $p = 2$. Moreover, fully discretized schemes based on the Fourier collocation method for spatial discretization and Strang splitting method for time discretization also preserve the uniform $L^2$-bound unconditionally.

  • Keywords

Allen-Cahn Equations, maximum principle, operator splitting, uniform $L^p$-bound, Fourier spectral methods.

  • AMS Subject Headings

65M70, 65R20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yangj7@sustc.edu.cn (Jiang Yang)

qd2125@columbia.edu (Qiang Du)

wzhang@csrc.ac.cn (Wei Zhang)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-213, author = {Yang , JiangDu , Qiang and Zhang , Wei}, title = {Uniform $L^p$-Bound of the Allen-Cahn Equation and Its Numerical Discretization}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {1-2}, pages = {213--227}, abstract = {

We study uniform bounds associated with the Allen-Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, shares the uniform $L^p$-bound for all even $p$, which also leads to the maximum principle. In comparison, a couple of other spatial discretization schemes, namely the Fourier spectral Galerkin method and spectral collocation method preserve the $L^p$-bound only for $p = 2$. Moreover, fully discretized schemes based on the Fourier collocation method for spatial discretization and Strang splitting method for time discretization also preserve the uniform $L^2$-bound unconditionally.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10564.html} }
TY - JOUR T1 - Uniform $L^p$-Bound of the Allen-Cahn Equation and Its Numerical Discretization AU - Yang , Jiang AU - Du , Qiang AU - Zhang , Wei JO - International Journal of Numerical Analysis and Modeling VL - 1-2 SP - 213 EP - 227 PY - 2018 DA - 2018/01 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10564.html KW - Allen-Cahn Equations, maximum principle, operator splitting, uniform $L^p$-bound, Fourier spectral methods. AB -

We study uniform bounds associated with the Allen-Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, shares the uniform $L^p$-bound for all even $p$, which also leads to the maximum principle. In comparison, a couple of other spatial discretization schemes, namely the Fourier spectral Galerkin method and spectral collocation method preserve the $L^p$-bound only for $p = 2$. Moreover, fully discretized schemes based on the Fourier collocation method for spatial discretization and Strang splitting method for time discretization also preserve the uniform $L^2$-bound unconditionally.

Yang , JiangDu , Qiang and Zhang , Wei. (2018). Uniform $L^p$-Bound of the Allen-Cahn Equation and Its Numerical Discretization. International Journal of Numerical Analysis and Modeling. 15 (1-2). 213-227. doi:
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