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This paper aims to investigate the numerical approximation of a generalsecond order parabolic stochastic partial differential equation(SPDE) driven bymultiplicative and additive noise. Our main interest is on such SPDEs where thenonlinear part is stronger than the linear part, usually called stochasticdominated transport equations. Most standard numerical schemes lose their goodstability properties on such equations, including the current linear implicitEuler method. We discretise the SPDE in space by the finite element method andpropose a new scheme in time appropriate for such equations, called stochasticRosenbrock-Type scheme, which is based on the local linearisation of thesemi-discrete problem obtained after space discretisation. We provide a strongconvergence of the new fully discrete scheme toward the exact solution formultiplicative and additive noise. Our convergence rates are in agreement withresults in the literature. Numerical experiments to sustain our theoreticalresults are provided.

Strong Convergence of a Stochastic Rosenbrock-type Scheme for the Finite  Element Discretization of Semilinear SPDEs Driven by Multiplicative and  Additive Noise