Our focus is on simulating the dynamics of non-interacting particles, which, under certain assumptions, can be formally described by the Dean-Kawasaki equation. The Dean-Kawasaki equation can be solved numerically using standard finite volume methods. However, the numerical approximation implicitly requires a sufficiently large number of particles to ensure the positivity of the solution and accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to scenarios where the density is low. The aim is to create a hybrid algorithm that switches from a finite volume discretization to a particle-based method when the particle density falls below a certain threshold. We develop criteria for determining this threshold by comparing higher-order statistics obtained from the finite volume method with particle simulations. We then demonstrate the use of the resulting criteria for dynamic adaptation in both two- and three-dimensional spatial settings.

翻译：本文聚焦于模拟非相互作用粒子的动力学行为。在特定假设下，该动力学可通过Dean-Kawasaki方程进行形式化描述。该方程可采用标准有限体积法进行数值求解。然而，数值近似隐式要求足够多的粒子数量，以确保解的正定性及随机通量的精确逼近。为应对这一挑战，我们将粒子系统的混合算法扩展至低密度场景。目标是构建一种混合算法，当粒子密度低于特定阈值时，能够从有限体积离散化切换至基于粒子的数值方法。通过比较有限体积法与粒子模拟获得的高阶统计量，我们建立了判定该阈值的准则。随后，我们在二维与三维空间构型中，展示了基于该准则的动态自适应算法的应用。