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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.

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order 1 of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.

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.

In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner-Ellis theorem, we prove that the observable BT = u(T ) T , T > 0 of the exact solution u is exponentially tight and satisfies an LDP on L 2 (0, π; C). Then, we present the LDPs for both {B M T }T >0 of the spatial discretization {u M } M ∈N and {B M N } N∈N of the full discretization {u M N } M,N∈N , whereNτ are the discrete approximations of BT . Further, we show that both the semi-discretization {u M } M ∈N and the full discretization {u M N } M,N∈N based on temporal symplectic schemes can weakly asymptotically preserve the LDP of {BT }T >0 . These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear Schrödinger equation, and first provide an effective approach to approximating the LDP rate function in infinite dimensional space based on the numerical discretizations.