Stability and Error Estimates with Optimal $\varepsilon$-Dependence for Strang Splitting of the Allen--Cahn Equation
Year: 2026
Type: preprint
Abstract: In this work, long-time Sobolev stability and error estimates with optimal $\varepsilon$-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 $\mathcal{O}(\varepsilon^{-r}+1)$, consistent with the corresponding Sobolev bounds of the exact solution. A second-order error estimate $\mathcal{O}(\varepsilon^2τ^2)$ is further established on any prescribed finite time interval, exhibiting quadratic dependence on $\varepsilon$. Numerical experiments are presented to verify the optimal $\varepsilon$-dependence of both the stability and error estimates.
Source: arXiv (Cornell University)
Authors: Luo, Rongfu, Quan, Chaoyu, Tan, Zhijun, Wu, Yanyao
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: National Natural Science Foundation of China, Sun Yat-sen University, Natural Science Foundation of Guangdong Province, Guangdong Province Key Laboratory of Computational Science, Shenzhen Science and Technology Innovation Program, Basic and Applied Basic Research Foundation of Guangdong Province
OpenAlex ID: W7221828798