Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation

TL;DR

This paper analyzes the convergence of a finite difference method for the stochastic Cahn--Hilliard equation, establishing strong convergence orders in space and time, and demonstrating density convergence of the numerical solution.

Contribution

It provides the first detailed convergence analysis of a finite difference scheme for the stochastic Cahn--Hilliard equation with multiplicative noise, including density convergence.

Findings

Spatial semi-discrete solution and its Malliavin derivative have strong order 1 convergence.

Density of the numerical solution converges in L^1 to the exact density.

Temporal convergence order is nearly 3/8 with an exponential Euler method.

Abstract

This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal H\"older continuity of the spatial semi-discrete numerical solution.