我们引入了一种新的碰撞动力学方程的微观/宏观分解，该分解自然地融入了精确的空间边界条件。其思想是将分布函数 $f$ 在其整个定义域内表示为边界适应的麦克斯韦分布（不是通常与 $f$ 相关的麦克斯韦分布）与一个剩余动力学部分之和。该麦克斯韦分布的定义使得其“入射”速度矩与分布函数的“入射”速度矩一致。这一策略的重要后果如下：i) 在微观/宏观模型中不需要人工边界条件，并且 $f$ 的精确边界条件自然地转化为模型的宏观部分。ii) 它提供了一类新的所谓“渐近保持”（AP）数值格式：这类格式对于所有固定的正克努森数 $\eps$ 与原动力学方程一致，并且如果 $\eps o 0$ 且数值参数固定，则这些格式退化为各种相应渐近流体或扩散模型的一致数值格式。这里，该策略不仅在物理域内部提供 AP 格式，而且在空间边界层中也提供 AP 格式。我们在单组输运方程的扩散极限情况下进行了数值测试，并表明我们的 AP 格式恢复了边界层以及理论边界值的良好近似，该理论边界值通常由所谓的 Chandrasekhar 函数计算得到。
英文摘要：
We introduce a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions. The idea is to write the distribution fonction $f$ in all its domain as the sum of a Maxwellian adapted to the boundary (which is not the usual Maxwellian associated with $f$) and a reminder kinetic part. This Maxwellian is defined such that its 'incoming' velocity moments coincide with the 'incoming' velocity moments of the distribution function. Important consequences of this strategy are the following. i) No artificial boundary condition is needed in the micro/macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. ii) It provides a new class of the so-called 'Asymptotic preserving' (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number $\eps$, and if $\eps \to 0 $ with fixed numerical parameters then these schemes degenerate into consistent numerical schemes for the various corresponding asymptotic fluid or diffusive models. Here, the strategy provides AP schemes not only inside the physical domain but also in the space boundary layers. We provide a numerical test in the case of a diffusion limit of the one-group transport equation, and show that our AP scheme recovers the boundary layer and a good approximation of the theoretical boundary value, which is usually computed from to the so-called Chandrasekhar function.
In this paper we propose coupling conditions for a kinetic two velocity model for vehicular traffic for junctions with diverging lanes. We consider cases with and without directional preferences and present corresponding kinetic coupling conditions. From this kinetic network model coupling conditions for a macroscopic traffic model are derived. We use an analysis of the layer equations at the junction in combination with a suitable matching procedure with half-Riemann problems for the macroscopic model. In this way classical coupling conditions for scalar conservation laws for traffic flow on …
In this paper we propose coupling conditions for a kinetic two velocity model for vehicular traffic on networks. These conditions are based on the consideration of the free space on the respective roads. The macroscopic limit of the kinetic relaxation system is a classical scalar conservation law for traffic flow. Similar to the asymptotic limit of boundary value problems for kinetic models, we consider here the limit of the full network problem including the coupling conditions at the nodes. An asymptotic analysis of the interface layers at the nodes and a matching procedure using half-Rieman…
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling
INRIA Rennes - Bretagne Atlantique（法国国家信息与自动化研究所雷恩-大西洋布列塔尼中心）；Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes（南特大学让·勒雷数学实验室，CNRS UMR 6629）；Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625, Université de Rennes 1（雷恩大学1号数学研究中心，CNRS UMR 6625）；ENS Rennes（雷恩高等师范学院）
An Asymptotic-Preserving Micro--Macro Scheme for Plasma Simulations in Quasi-Neutral and Low-Mach-Number Regimes with Kinetic Upgrades
Harbin Institute of Technology（哈尔滨工业大学）；Université de Toulouse（图卢兹大学）；UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, CNRS, Institut de Mathématiques de Toulouse UMR 5219
This paper presents a hybrid numerical method for linear collisional kinetic equations with diffusive scaling. The aim of the method is to reduce the computational cost of kinetic equations by taking advantage of the lower dimensionality of the asymptotic fluid model while reducing the error induced by the latter approach. It relies on two criteria motivated by a pertubative approach to obtain a dynamic domain decomposition. The first criterion quantifies how far from a local equilibrium in velocity the distribution function of particles is. The second one depends only on the macroscopic quant…
We introduce a new micro-macro Markov chain Monte Carlo method (mM-MCMC) with indirect reconstruction to sample invariant distributions of molecular dynamics systems that exhibit a time-scale separation between the microscopic (fast) dynamics, and the macroscopic (slow) dynamics of some low-dimensional set of reaction coordinates. The algorithm enhances exploration of the state space in the presence of metastability by allowing larger proposal moves at the macroscopic level, on which a conditional accept-reject procedure is applied. Only when the macroscopic proposal is accepted, the full micr…
In this paper, authors focus effort on improving the conventional discrete velocity method (DVM) into a multiscale scheme in finite volume framework for gas flow in all flow regimes. Unlike the typical multiscale kinetic methods unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme (DUGKS), which concentrate on the evolution of the distribution function at the cell interface, in the present scheme the flux for macroscopic variables is split into the equilibrium part and the nonequilibrium part, and the nonequilibrium flux is calculated by integrating the discrete distributi…
We introduce a generalised micro-macro Markov chain Monte Carlo (mM-MCMC) method with pseudo-marginal approximation to the free energy, that is able to accelerate sampling of the microscopic Gibbs distributions when there is a time-scale separation between the macroscopic dynamics of a reaction coordinate and the remaining microscopic degrees of freedom. The mM-MCMC method attains this efficiency by iterating four steps: i) Propose a new value of the reaction coordinate; ii) Accept or reject the macroscopic sample; iii) Run a biased simulation that creates a microscopic molecular instance that…
