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

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 $\frac{3}{8}$, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution.

References in corpus (3)

• Finite Element Approximation of the Cahn-Hilliard-Cook equation
• Finite element approximation of the linearized Cahn-Hilliard-Cook equation
• On strongly Petrovskii's parabolic SPDEs in arbitrary dimension and the stochastic Cahn-Hilliard equation

Cited by in corpus (2)

• Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise
• Asymptotics of large deviations of finite difference method for stochastic Cahn--Hilliard equation