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

We present numerical schemes to integrate stochastic partial differential equations which describe the spatio-temporal dynamics of reaction-diffusion problems under the effect of internal fluctuations. The schemes conserve the non-negativity 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 schemes to two different aspects of the Reggeon model, namely, the study of its nonequilibrium phase transition and the dynamics of fluctuating pulled fronts. In the latter case, our approach allows us to reproduce microscopic properties quantitatively within the continuum model.

Figure 1

Conditional probability density as a function of ${\rho }_{t+\Delta t}$ for Eq. (2) with $\Delta t=0.1$, ${\rho }_t=1$ (left), and ${\rho }_t={10}^{-2}$ (right). Solid lines are the exact solution from Eq. (5), while dashed lines are the approximations obtained using the Euler scheme (3).

Figure 2

Left: convergence analysis for the different algorithms. Points are the critical value of $\sigma$ as a function $\Delta t$ (below) and of $\Delta t^{1∕2}$ (up) while dashed lines are linear fits to the data. The system size is $L=400$. Right: time dependence of the mean density $\overline{\rho }\left(t\right)$ at the critical point $\sigma ={\sigma }_c\left(\Delta t\right)$ with $\Delta t={10}^{-2}$ obtained using the different algorithms (lines are shifted vertically for clarity). The thin line is the power law $\overline{\rho }\left(t\right)\sim t^{-\delta }$ with $\delta =0.1595$. The system size is $L=1000$.

Figure 3

Front diffusion coefficient as a function of $\sigma$ for the different algorithms and different $\Delta t$, compared with hybrid MC simulations of the microscopic model $A\leftrightarrow A+A$ [14]. The solid (dashed) line is the ${\log }^{-3}N$ $\left({\log }^{-6}N\right)$ power law.