Stochastic exponential integrators for a finite element discretisation of SPDEs with additive noise

We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square L² norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.