  We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDEs) driven
by space-time noise, for multiplicative and additive noise. We examine
convergence of exponential integrators for multiplicative and additive noise.
We consider noise that is in trace class and give a convergence proof in the
mean square $L^{2}$ norm. We discretize in space with the finite element method
and in our implementation we examine both the finite element and the finite
volume methods. We present results for a linear reaction diffusion equation in
two dimensions as well as a nonlinear example of two-dimensional stochastic
advection diffusion reaction equation motivated from realistic porous media
flow.
