A discontinuous Galerkin approximation of the Dean--Kawasaki equation
We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean--Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.