A step towards holistic discretisation of stochastic partial differential equations

TL;DR

Problem: developing reliable, efficient discretisations for noisy, dissipative SPDEs that preserve subgrid-scale stochastic effects. Approach: apply stochastic centre manifold theory on a small domain to derive a one-element model, then use a normal-form reduction to remove fast convolutions and obtain a weak amplitude equation with effective noise. Key contributions: explicit amplitude equations showing stochastic resonance induces drift and new stochastic terms, including a drift term of $-\frac{\sigma^2}{88}$ that shifts the stability threshold to $\gamma > -\frac{\sigma^2}{88}$, enabling large time-step simulations; and a framework to replace fast-time convolutions with effective noises via long-time statistics. Significance: provides a practical, theory-backed route to holistic SPDE discretisation balancing fidelity to subgrid stochastic dynamics with computational efficiency; further theoretical development is needed.

Abstract

The long term aim is to use modern dynamical systems theory to derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider a small domain and apply stochastic centre manifold techniques to derive a model. The approach automatically parametrises subgrid scale processes induced by spatially distributed stochastic noise. It is important to discretise stochastic partial differential equations carefully, as we do here, because of the sometimes subtle effects of noise processes. In particular we see how stochastic resonance effectively extracts new noise processes for the model which in this example helps stabilise the zero solution.

Figures (5)

• Figure 1: numerical solution over time $0<t<3$ of the spde (\ref{['eq:oburgnm']}) on the domain $0<x<\pi$ with stochastic forcing (\ref{['eq:onoise']}) truncated to the first seven spatial modes. Parameters: $\gamma=0$ so $u\propto\sin x$ is linearly neutral although nonlinearly stable; $\sigma=1$ for large forcing; numerically $\Delta x=\pi/16$ and $\Delta t=0.01$ .
• Figure 2: numerical solution of the sde model (\ref{['eq:oomod']}) with small, $\sigma=0.5$, and large, $\sigma=2$, noise. The amplitude $a$ of the $\sin x$ mode decays for large noise, but not for small. Parameters: $\gamma=-0.03$ to promote linear growth of $a$, and $\Delta t=0.1$ .
• Figure 3: numerical solution of the spde (\ref{['eq:oburgnm']}) with relatively weak noise limited to just $\phi=\phi_2(t)\sin 2x$ showing convergence to a nonlinearly stabilised $\sin x$ mode that is perturbed by the noise. Parameters: $\sigma=0.5$ is small, $\gamma=-0.03$ to generate linear growth of the $\sin x$ mode, $\Delta t=0.05$ and $\Delta x=\pi/8$ .
• Figure 4: numerical solution of the spde (\ref{['eq:oburgnm']}) with strong noise limited to just $\phi=\phi_2(t)\sin 2x$ showing the $\sin x$ mode decays. Parameters: $\sigma=2$, $\gamma=-0.03$ to promote linear growth of the $\sin x$ mode, $\Delta t=0.05$ and $\Delta x=\pi/8$ .
• Figure 5: simulations of the long time model (\ref{['eq:oomodl']}) for small, $\sigma=0.5$, and large, $\sigma=2$, noise over long times. Parameters: $\Delta t=1$, $\gamma=-0.03$ .