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

TL;DR

This paper establishes the weak intermittency and second moment bounds of a fully discrete numerical scheme for the stochastic heat equation with multiplicative noise, ensuring the scheme captures key intrinsic properties of the exact solution.

Contribution

The paper introduces a fully discrete finite difference scheme with a $ heta$-method that preserves the exponential growth order of the second moment of the stochastic heat equation.

Findings

Numerical second moments grow at least as $ ext{exp}igrace C ext{lambda}^2 tigrace$ and at most as $ ext{exp}igrace C ext{lambda}^4 tigrace$.

The scheme preserves the sharp exponential order $C ext{lambda}^4 t$ of the second moment for large spatial partitions.

The numerical solutions exhibit weak intermittency similar to the exact solution.

Abstract

In this paper, we first prove the weak intermittency, and in particular the sharp exponential order $C\lambda^4t$ of the second moment of the exact solution of the stochastic heat equation with multiplicative noise and periodic boundary condition, where $\lambda>0$ 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 $\theta$-scheme. We prove that the second moment of numerical solutions of both spatially semi-discrete and fully discrete schemes grows at least as $\exp\{C\lambda^2t\}$ and at most as $\exp\{C\lambda^4t\}$ 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 $C\lambda^4t$ of the second moment of the exact solution for large spatial partition number.