  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.
