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

Wansheng Wang and Yi Huang

Mathematics and Computers in Simulation (MATCOM), 2023, vol. 207, issue C, 80-96

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.

Keywords: Fractional Laplacian operator; Space-fractional Allen–Cahn equation; Backward Euler method; Dissipativity; Global attractor (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:207:y:2023:i:c:p:80-96

DOI: 10.1016/j.matcom.2022.12.012

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