  We present new numerical schemes to integrate stochastic partial differential
equations which describe the spatio-temporal dynamics of reaction-diffusion
(RD) problems under the effect of internal fluctuations. The schemes conserve
the nonnegativity of the solutions and incorporate the Poissonian nature of
internal fluctuations at small densities, their performance being limited by
the level of approximation of density fluctuations at small scales. We apply
the new schemes to two different aspects of the Reggeon model namely, the study
of its non-equilibrium phase transition and the dynamics of fluctuating pulled
fronts. In the latter case, our approach allows to reproduce quantitatively for
the first time microscopic properties within the continuum model.
