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Cai, Yao-Yuan; Sun, Hai-Wei; Tam, Sik-Chung

Numerical study of a fast two-level Strang splitting method for spatial fractional Allen-Cahn equations. (English) Zbl 07698931

J. Sci. Comput. 95, No. 3, Paper No. 71, 23 p. (2023).
Summary: In this paper, a numerical method to solve the multi-dimensional spatial fractional Allen-Cahn equations has been investigated. After semi-discretizating the equations, a system of nonlinear ordinary differential equations with a Toeplitz structure is induced. We propose to split the Toeplitz matrix into the sum of a circulant matrix and a skew-circulant matrix, and apply the Strang splitting method. Such a two-level Strang splitting method will reduce the computational complexity to \({\mathcal{O}}(q\log q)\). Moreover, it preserves not only the discrete maximum principle unconditionally but also second-order convergence as well. By introducing a new modified energy formula, the energy dissipation property can be guaranteed. Finally, some numerical experiments are conducted to confirm the theories we put forward.

Cited in 1 Document

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35Rxx Miscellaneous topics in partial differential equations

Keywords:

altered two-level Strang splitting method; circulant and skew-circulant matrix; fast Fourier transform; discrete maximum principle; modified energy decay

Software:

mftoolbox
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References:

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