  In this paper, we investigate a numerical approximation of a general second
order semilinear parabolic non-autonomous stochastic partial differential
equation (SPDE) driven by additive noise. Numerical approximations for
autonomous SPDEs are thoroughly investigated in the literature while the
non-autonomous case is not yet well understood.
  We discretize the non-autonomous SPDE in space by the finite element method
and in time by the Magnus-type integrator. We provide a strong convergence
proof of the fully discrete scheme toward the mild solution in the
root-mean-square $L^2$ norm. Appropriate assumptions on the drift term and the
noise allow to achieve optimal convergence order in time greater than $1/2$,
without any logarithmic reduction of convergence order in time. In particular,
for trace class noise, we achieve optimal convergence orders
$\mathcal{O}\left(h^{2-\epsilon}+\Delta t\right)$, where $\epsilon$ is a
positive number small enough. Numerical simulations are provided to illustrate
our theoretical results.
