Finite difference method for stochastic Cahn–Hilliard equation: Strong convergence rate and density convergence
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This paper presents the strong convergence rate and density convergence of a spatial finite difference method (FDM) when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither global Lipschitz nor one-sided Lipschitz. To handle this difficulty, we first utilize an interpolation approach to derive the discrete H^1-regularity of the numerical solution. This is the key to deriving the optimal strong convergence order 1 of the numerical solution. Further, we propose a novel localization argument to estimate the total variation distance between the exact and numerical solutions, which along with the existence of the density of the numerical solution finally yields the convergence of density in L^1(ℝ) of the numerical solution. This partially answers positively to the open problem emerged in [9,Section 5] on computing the density of the exact solution numerically.
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