Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise

TL;DR

This paper introduces a novel explicit numerical scheme for the stochastic Cahn--Hilliard equation with additive noise, demonstrating strong convergence and improved computational efficiency over implicit methods.

Contribution

It presents the first explicit scheme for this equation, combining spectral Galerkin and tamed exponential Euler methods, with proven strong convergence rates.

Findings

The explicit scheme converges strongly to the exact solution.

Numerical experiments confirm theoretical convergence rates.

The scheme offers computational efficiency advantages.

Abstract

In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn--Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. Different from the tamed time-stepping schemes for stochastic Allen--Cahn equations, essential difficulties arise in the analysis due to the presence of the unbounded linear operator in front of the nonlinearity. To overcome them, new and non-trivial arguments are developed in the present work. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn--Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.