Dean--Kawasaki 方程的不连续伽辽金逼近
A discontinuous Galerkin approximation of the Dean--Kawasaki equation
- University of Bath（巴斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出并分析了一种任意阶不连续伽辽金方法求解 Dean--Kawasaki 方程，通过保持离散噪声交叉变差结构，获得粒子系统涨落的 O(h^p) 弱误差估计，并适用于非结构化网格。
AI中文摘要：
我们针对 Dean--Kawasaki 方程引入并分析了一种任意阶空间不连续伽辽金（dG）方法。该方程是一个高度奇异的随机偏微分方程，用于模拟在大粒子数 N ≫ 1 情况下 N 个扩散粒子的密度涨落。我们的出发点是：在有限元空间上离散乘性、散度形式的噪声，同时保持其在离散层面的交叉变差结构的一般过程。该构造是显式的、逐单元的，适用于连续空间和不连续空间，并可推广到一般的迁移率。利用该方法，我们证明了半离散格式的涨落与底层粒子系统的涨落之间的弱误差估计，阶数为 O(h^p)，同时还有一个在尺度区域 Nh^d ≫ 1 中指数小的修正项，该修正项源于格式不保持正性。所得方法在局部和全局上都是守恒的，并适用于非结构化单纯形网格。定量数值实验支持了分析结果，进一步的实验则展示了该方法在理论范围之外的应用：指示函数观测、外部势和相互作用势、对流主导区域以及反射边界条件。
英文摘要：
We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean--Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.
