A High-Order Numerical Scheme for Allen-Cahn Equation via Strang Splitting and Fourier Spectral Method

Abstract

This paper proposes a practical numerical scheme for solving the Allen-Cahn (AC) equation by integrating Strang operator splitting, the Fourier spectral method, and SSP-RK (strong stability-preserving Runge-Kutta) time integration. This approach enhances the stability of phase-field simulations while ensuring computational efficiency by decoupling the treatment of stiff nonlinear terms and linear diffusion terms: the nonlinear reaction term is explicitly advanced using SSP-RK, leveraging its strong stability-preserving properties to control solution boundedness under moderate time step sizes; the linear diffusion term is solved exactly via Fourier spectral differentiation, significantly reducing the iterative cost of traditional implicit schemes. Spatial discretization employs the spectral method, achieving superior interface resolution compared to finite differences in regular domains. Numerical verifications demonstrate that: under a moderate stiffness parameter ( ), when the time step size is refined to 0.00625, the error stably decreases from an initial  to , with convergence properties approaching the theoretical second-order accuracy; under strong stiffness regimes ( ), the scheme maintains effective convergence trends, with error reduction magnitudes reaching an order of magnitude. Physical property verifications confirm that the scheme strictly preserves the maximum principle (phase-field values always lie within the [0,1] interval) under two/three-dimensional random initial conditions, and the discrete energy exhibits monotonic decay characteristics consistent with the evolutionary laws of the double-well free energy. Computational efficiency tests show that, for moderate-scale problems, the scheme saves approximately 30–40% of computational time compared to standard implicit schemes. This scheme provides a new option balancing accuracy and efficiency for phase-field modeling, though its strong stiffness adaptability and large-scale scalability still require in-depth study.