Model subgrid microscale interactions to accurately discretise stochastic partial differential equations

Constructing discrete models of stochastic partial differential equations is very delicate. Here we describe how stochastic centre manifold theory provides novel support for spatial discretisations of the nonlinear advection-diffusion dynamics of the stochastically forced Burgers’ equation. Dividing the physical domain into finite sized elements empowers the approach to resolve fully coupled dynamical interactions between neighbouring elements through its theoretical support. The crucial aspect of this work is that the underlying theory organises how we may deal with the multitude of subgrid microscale noise processes interacting via the nonlinear dynamics within and between neighbouring elements to affect macroscale dynamics. 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. Nonlinear interactions have two further consequences: additive forcing generates multiplicative noise effects in the discretisation; and effectively new noise processes appear in the macroscale discretisation. The techniques and theory developed here may be applied to discretise many dissipative stochastic partial differential equations.