A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle

TL;DR

This paper introduces a fast, two-level Strang splitting method for multi-dimensional fractional Allen-Cahn equations that preserves the maximum principle and achieves second-order accuracy, significantly reducing computational complexity.

Contribution

The paper develops a novel splitting method that efficiently solves multi-dimensional fractional Allen-Cahn equations using FFT and preserves the maximum principle.

Findings

Method is unconditionally stable and preserves the maximum principle.

Achieves second-order accuracy in both time and space.

Numerical tests confirm efficiency and theoretical results.

Abstract

In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to calculate the expensive Toeplitz matrix exponential. Theoretically, the discrete maximum principle of our method is unconditionally preserved. Moreover, the analysis of error in the infinite norm with second-order accuracy is conducted in both time and space. Finally, numerical tests are given to corroborate our theoretical conclusions and the efficiency of the proposed method.