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.

翻译：正则化惯性Dean-Kawasaki模型（RIDK）——由作者与J. Zimmer在先前工作中提出——是一类非线性随机偏微分方程，用于捕捉大规模粒子系统中粒子密度与动量密度围绕平均场极限的涨落。我们聚焦以下两个方面。首先，我们建立了RIDK模型的间断Galerkin（DG）离散格式：给出了与RIDK模型波动性质一致的网格单元界面数值通量定义，确保了模拟稳定性，并量化了均方意义下收敛至连续RIDK模型的收敛速率。其次，我们引入RIDK模型的修正项以保证密度正性（该性质仅以"高概率意义"适用于原始RIDK模型）。通过数值模拟，我们证明修正项能产生物理真实且保持正性的密度分布。在附加正则性约束的特定情形下，我们同时证明了正性。最后，我们将该方法应用于扩散反应粒子系统。我们的Python代码以开源形式提供。