  We consider the discretization in time of a system of parabolic stochastic
partial differential equations with slow and fast components; the fast equation
is driven by an additive space-time white noise. The numerical method is
inspired by the Averaging Principle satisfied by this system, and fits to the
framework of Heterogeneous Multiscale Methods.The slow and the fast components
are approximated with two coupled numerical semi-implicit Euler schemes
depending on two different timestep sizes. We derive bounds of the
approximation error on the slow component in the strong sense - approximation
of trajectories - and in the weak sense - approximation of the laws. The
estimates generalize the results of \cite{E-L-V} in the case of infinite
dimensional processes.
