  This paper aims to investigate the numerical approximation of a general
second order parabolic stochastic partial differential equation(SPDE) driven by
multiplicative and additive noise. Our main interest is on such SPDEs where the
nonlinear part is stronger than the linear part, usually called stochastic
dominated transport equations. Most standard numerical schemes lose their good
stability properties on such equations, including the current linear implicit
Euler method. We discretise the SPDE in space by the finite element method and
propose a new scheme in time appropriate for such equations, called stochastic
Rosenbrock-Type scheme, which is based on the local linearisation of the
semi-discrete problem obtained after space discretisation. We provide a strong
convergence of the new fully discrete scheme toward the exact solution for
multiplicative and additive noise. Our convergence rates are in agreement with
results in the literature. Numerical experiments to sustain our theoretical
results are provided.
