An explicit scheme for stochastic Allen-Cahn equations with space-time white noise near the sharp interface limit

TL;DR

This paper develops an explicit exponential integrator for stochastic Allen-Cahn equations with space-time white noise, achieving improved error bounds and milder time step restrictions near the sharp interface limit.

Contribution

It introduces a novel explicit scheme with uniform-in-time and epsilon bounds, reducing restrictions on time step size and improving convergence analysis for SPDEs with low regularity.

Findings

Convergence in total variation distance with polynomial dependence on epsilon and T.

Mild, epsilon-independent time step restriction achieved.

Numerical experiments confirm theoretical convergence and interface capturing.

Abstract

This article investigates time-discrete approximations of Allen-Cahn type SPDEs driven by space-time white noise near the sharp interface limit $\epsilon\to 0$, where the small parameter $\epsilon$ is the diffuse interface thickness. We propose an explicit and easily implementable exponential integrator with a modified nonlinearity for the considered problem. Uniform-in-time and uniform-in-$\epsilon$ moment bounds of the scheme are established and the convergence in total variation distance of order $O(T\cdot\text{Poly}(\epsilon^{-1})\tau^\gamma),\gamma<\tfrac12$ is established, between the law of the numerical scheme and that of the SPDE over $[0,T]$. In contrast to the exponential dependence due to standard arguments, the obtained error bound depends on $\epsilon^{-1}$ and $T$ polynomially. By incorporating carefully chosen method parameters, we only require a mild and $\epsilon$-independent restriction on the time step-size $\tau$, getting rid of the severe restriction $\tau =O(\epsilon^\sigma),\sigma\geq1$ in the literature. Also, a uniform-in-time error bound of order $O(\tau^\gamma),\gamma<\tfrac12$, is obtained for a fixed $\epsilon=1$, which improves the existing ones in the literature and matches the classical weak convergence rate in the globally Lipschitz setting. The error analysis is highly nontrivial due to the low regularity of the considered problem, the super-linear growth of the drift, the non-smooth observables inherent in the total variation metric and the presence of the small interface parameter $\epsilon\to0$. These difficulties are addressed by introducing a new strategy of nonlinearity modification and establishing refined regularity estimates for the associated Kolmogorov equation to an auxiliary process with non-smooth test functions. Numerical experiments confirm the theoretical convergence and the ability of interface-capturing for the proposed scheme.