Stability and Error Estimates with Optimal $\epsilon$-Dependence for Strang Splitting of the Allen--Cahn Equation

Abstract

In this work, long-time Sobolev stability and error estimates with optimal $\epsilon$-dependence are established for the Strang splitting method applied to the Allen--Cahn equation with periodic boundary conditions. By combining the modified energy dissipation law, and Hölder-Gagliardo-Nirenberg product estimates, the numerical solution is proven to admit uniform-in-time $H^r$-seminorm bounds $\mathcal{O}({\epsilon }^{-r}+1)$, consistent with the corresponding Sobolev bounds of the exact solution. A second-order error estimate $\mathcal{O}({\epsilon }^2{\tau }^2)$ is further established on any prescribed finite time interval, exhibiting quadratic dependence on $\epsilon$. Numerical experiments are presented to verify the optimal $\epsilon$-dependence of both the stability and error estimates.