  In this paper, we consider a new approach for semi-discretization in time and
spatial discretization of a class of semi-linear stochastic partial
differential equations (SPDEs) with multiplicative noise. The drift term of the
SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the
diffusion term is assumed to be globally Lipschitz continuous. Our new strategy
for time discretization is based on the Milstein method from stochastic
differential equations. We use the energy method for its error analysis and
show a strong convergence order of nearly $1$ for the approximate solution. The
proof is based on new H\"older continuity estimates of the SPDE solution and
the nonlinear term. For the general polynomial-type drift term, there are
difficulties in deriving even the stability of the numerical solutions. We
propose an interpolation-based finite element method for spatial discretization
to overcome the difficulties. Then we obtain $H^1$ stability, higher moment
$H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results
using numerical and stochastic techniques. The nearly optimal convergence
orders in time and space are hence obtained by coupling all previous results.
Numerical experiments are presented to implement the proposed numerical scheme
and to validate the theoretical results.
