  In this paper, we consider the numerical approximation of a general second
order semilinear 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 also called stochastic reactive
dominated transport equations. Most numerical techniques, including current
stochastic exponential integrators lose their good stability properties on such
equations. Using finite element for space discretization, we propose a new
scheme appropriated on such equations, called stochastic exponential Rosenbrock
scheme (SERS) based on local linearization at every time step of the
semi-discrete equation obtained after space discretization. We consider noise
that is in a trace class and give a strong convergence proof of the new scheme
toward the exact solution in the root-mean-square $L^2$ norm. Numerical
experiments to sustain theoretical results are provided.
