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The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by theauthors and J. Zimmer in earlier works -- is a nonlinear stochastic PDEcapturing fluctuations around the mean-field limit for large-scale particlesystems in both particle density and momentum density. We focus on thefollowing two aspects. Firstly, we set up a Discontinuous Galerkin (DG)discretisation scheme for the RIDK model: we provide suitable definitions ofnumerical fluxes at the interface of the mesh elements which are consistentwith the wave-type nature of the RIDK model and grant stability of thesimulations, and we quantify the rate of convergence in mean square to thecontinuous RIDK model. Secondly, we introduce modifications of the RIDK modelin order to preserve positivity of the density (such a feature only holds in a''high-probability sense'' for the original RIDK model). By means of numericalsimulations, we show that the modifications lead to physically realistic andpositive density profiles. In one case, subject to additional regularityconstraints, we also prove positivity. Finally, we present an application ofour methodology to a system of diffusing and reacting particles. Our Pythoncode is available in open-source format.

The Regularised Inertial Dean-Kawasaki equation: discontinuous Galerkin approximation and modelling for low-density regime