随机heat equation全离散格式的弱收敛性和二阶矩界
Weak intermittency and second moment bound of a fully discrete scheme for stochastic heat equation
摘要：
本文首先证明了带乘性噪声和周期边界条件的随机热方程精确解的弱非线性，特别是二阶矩的尖指数阶$Cλ^4t$，其中$λ>0$表示噪声的水平.为了在数值上继承原方程的这些内在性质，我们引入了一个全离散格式，其空间方向基于有限差分法，时间方向基于$θ$格式。我们证明了在自然条件下，对于大的$t$，半离散和全离散格式的数值解的二阶矩至少增长为$\exp\{Cλ^2t\}$，最多增长为$\exp\{Cλ^4t\}$，这意味着这些数值解的弱连续性.此外，一个更新的方法被应用到表明，这两个数值格式可以保持尖锐的指数阶$Cλ^4t$的精确解的二阶矩的大空间分区数。
英文摘要：
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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