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

02/18/2022

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by   Jialin Hong, et al.

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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(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 3/8, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution.