Numerical schemes for continuum models of reaction-diffusion systems subject to internal noise

TL;DR

This paper introduces new numerical schemes for stochastic reaction-diffusion equations that preserve nonnegativity and accurately model internal noise, enabling detailed study of phase transitions and front dynamics.

Contribution

The paper develops novel numerical schemes that handle internal fluctuations in reaction-diffusion models while maintaining solution positivity and capturing microscopic properties.

Findings

Successfully simulate non-equilibrium phase transitions.

Reproduce microscopic properties of fluctuating fronts.

Enhance understanding of internal noise effects in continuum models.

Abstract

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.