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Analytical and numerical dissipativity for the space-fractional Allen–Cahn equation

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  • Wang, Wansheng
  • Huang, Yi

Abstract

This paper is concerned with the analytical and numerical dissipativity of the space-fractional Allen–Cahn equation, a generalization of the classic Allen–Cahn equation by replacing the local Laplacian with a nonlocal fractional Laplacian. It is first proved that the continuous dynamical system is dissipative as its local counterpart in Hα and Lq, q=2k+2 for k≥0, spaces. Then it is shown that the backward Euler method preserves the dissipativity of the underlying system, that is, the discrete-in-time dynamical system with time-step parameter τ is still dissipative in Hα and Lq spaces. The existence of the global attractor for both continuous and discrete dynamical systems are then obtained. A numerical example is given to confirm the theoretical results.

Suggested Citation

  • Wang, Wansheng & Huang, Yi, 2023. "Analytical and numerical dissipativity for the space-fractional Allen–Cahn equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 80-96.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:80-96
    DOI: 10.1016/j.matcom.2022.12.012
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    References listed on IDEAS

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    2. James W. Kirchner & Xiahong Feng & Colin Neal, 2000. "Fractal stream chemistry and its implications for contaminant transport in catchments," Nature, Nature, vol. 403(6769), pages 524-527, February.
    3. Ainsworth, Mark & Mao, Zhiping, 2017. "Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 264-273.
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