Abstract. We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of the noise and the approximation of the exponential function by a rational fraction, we introduce a new scheme, designed for finite elements, finite volumes or finite differences space discretization, similar to the schemes in [4,7] for spectral methods and [1] for finite element methods. We use the projection operator, the smoothing effect of the positive definite selfadjoint operator and linear functionals of the noise in Fourier space to obtain higher order approximations. We consider noise that is white in time and either in H 1 or H 2 in space and give convergence proofs in the mean square L 2 norm for a diffusion reaction equation and in mean square H 1 norm in the presence of an advection term. For the exponential integrator we rely on computing the exponential of a non-diagonal matrix. In our numerical results we use two different efficient techniques: the real fast Léja points and Krylov subspace techniques. 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.