Resolve the multitude of microscale interactions to holistically discretise the stochastically forced Burgers' partial differential equation

Abstract

[Abstract]: Constructing discrete models of stochastic partial differential equations is very delicate. Here we use modern dynamical systems theory to derive spatial discretisations of the nonlinear advection-diffusion dynamics of the stochastically forced Burgers' partial differential equation. In a region of the domain far from any spatial boundaries, stochastic centre manifold theory supports a discrete model for the dynamics. The trick to the application of the theory is to divide the physical domain into finite sized elements by introducing insulating internal boundaries which are later removed to fully couple the dynamical interactions between neighbouring elements. Burgers' equation is used as an example. The approach automatically parametrises the microscale, subgrid structures within each element induced by spatially varying stochastic forcing. The crucial aspect of this work is that we explore how a multitude of noise processes interact via the nonlinear dynamics within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in the spatial discretisation. Further, the nonlinearity in the dynamics has two consequences: the example additive forcing generates multiplicative noise effects in the discretisation; and effectively new noise sources are abstracted over the macroscopic time scales resolved by the discretisation. The techniques and theory developed here may be applied to discretise many dissipative stochastic partial differential equations.