Stability and Convergence of Strang Splitting Method for the Allen-Cahn Equation with Homogeneous Neumann Boundary Condition

2025/04/10 by Quan, Chaoyu, Tan, Zhijun, Wu, Yanyao
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2504.07520

Abstract

The Strang splitting method has been widely used to solve nonlinear reaction-diffusion equations, with most theoretical convergence analysis assuming periodic boundary conditions. However, such analysis presents additional challenges for the case of homogeneous Neumann boundary condition. In this work the Strang splitting method with variable time steps is investigated for solving the Allen--Cahn equation with homogeneous Neumann boundary conditions. Uniform H^k-norm stability is established under the assumption that the initial condition u⁰ belongs to the Sobolev space H^k(Ω) with integer k≥ 0, using the Gagliardo--Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the H^k-norm for initial conditions u⁰ ∈ H^k+6(Ω), based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.

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• Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation
• Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
• Strang splitting method for semilinear parabolic problems with\n inhomogeneous boundary conditions: a correction based on the flow of the\n nonlinearity
• An initial-boundary corrected splitting method for diffusion-reaction problems