Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation
Provides rigorous convergence analysis for a numerical method applied to a stochastic PDE, which is incremental for researchers in numerical analysis of SPDEs.
The paper proves that the spatial finite difference method for the stochastic Cahn-Hilliard equation achieves strong convergence order 1, and the density of the semi-discrete solution converges in L^1. Temporal discretization with exponential Euler yields nearly 3/8 strong convergence order.
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 Hölder continuity of the spatial semi-discrete numerical solution.