# Weak intermittency and second moment bound of a fully discrete scheme for stochastic heat equation

In this paper, we first prove the weak intermittency, and in particular the sharp exponential order Cλ^4t of the second moment of the exact solution of the stochastic heat equation with multiplicative noise and periodic boundary condition, where λ>0 denotes the level of the noise. In order to inherit numerically these intrinsic properties of the original equation, we introduce a fully discrete scheme, whose spatial direction is based on the finite difference method and temporal direction is based on the θ-scheme. We prove that the second moment of numerical solutions of both spatially semi-discrete and fully discrete schemes grows at least as exp{Cλ^2t} and at most as exp{Cλ^4t} for large t under natural conditions, which implies the weak intermittency of these numerical solutions. Moreover, a renewal approach is applied to show that both of the numerical schemes could preserve the sharp exponential order Cλ^4t of the second moment of the exact solution for large spatial partition number.

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