Numerical Study of a Fast Two-Level Strang Splitting Method for Spatial Fractional Allen-Cahn Equations

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 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.

Automated summary

This paper investigates a numerical method for solving the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretizing the equations, a system of nonlinear ordinary differential equations with a Toeplitz structure is obtained. The author proposes a two-level Strang splitting method by splitting the Toeplitz matrix into a circulant matrix and a skew-circulant matrix to reduce computational complexity. This method unconditionally preserves the discrete maximum principle and achieves second-order convergence. Numerical experiments are conducted to validate the proposed theories.