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

The Regularised Inertial Dean-Kawasaki model (RIDK) is a nonlinear stochastic PDE which captures fluctuations around the mean-field limit for large-scale particle systems in both density and momentum. 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 does not hold 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.