论文标题
正则惯性院长卡瓦萨基方程：低密度状态的不连续的盖尔金近似和建模
The Regularised Inertial Dean-Kawasaki equation: discontinuous Galerkin approximation and modelling for low-density regime
论文作者
论文摘要
由作者和较早作品的J. Zimmer引入的正规惯性院长卡瓦萨基模型（RIDK）是一种非线性随机PDE，可在粒子密度和动量密度的大规模粒子系统的平均场限制周围捕获波动。我们专注于以下两个方面。首先，我们为RIDK模型设置了一个不连续的Galerkin（DG）离散方案：我们在网格元素的接口上提供了适当的数值通量定义，这些定义与RIDK模型的波型和授予稳定性一致的网格元素的界面，并量化了模拟的均值稳定性，并且我们量化了平均平均平均平均值速率的连续骑行模型。其次，我们介绍了RIDK模型的修改，以保持密度的阳性（这种功能仅在原始RIDK模型中具有“高概率意义”）。通过数值模拟，我们表明修饰导致物理逼真和正密度曲线。在一种情况下，受到额外的规律性约束，我们也证明了积极性。最后，我们提出了我们的方法的应用于扩散和反应粒子系统。我们的Python代码以开源格式获得。
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.
