  The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by the
authors and J. Zimmer in earlier works -- is a nonlinear stochastic PDE
capturing fluctuations around the mean-field limit for large-scale particle
systems in both particle density and momentum density. We focus on the
following two aspects. Firstly, we set up a Discontinuous Galerkin (DG)
discretisation scheme for the RIDK model: we provide suitable definitions of
numerical fluxes at the interface of the mesh elements which are consistent
with the wave-type nature of the RIDK model and grant stability of the
simulations, and we quantify the rate of convergence in mean square to the
continuous RIDK model. Secondly, we introduce modifications of the RIDK model
in 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 numerical
simulations, we show that the modifications lead to physically realistic and
positive density profiles. In one case, subject to additional regularity
constraints, we also prove positivity. Finally, we present an application of
our methodology to a system of diffusing and reacting particles. Our Python
code is available in open-source format.
