This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to ϕ-divergences and a corresponding closure procedure based on minimization of ϕ-divergences. The proposed description encapsulates as special cases Grad's classical closure based on expansion in Hermite polynomials and Levermore's entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.
Direct simulation of physical processes on a kinetic level is prohibitively expensive in aerospace applications due to the extremely high dimension of the solution spaces. In this paper, we consider the moment system of the Boltzmann equation, which projects the kinetic physics onto the hydrodynamic scale. The unclosed moment system can be solved in conjunction with the entropy closure strategy. Using an entropy closure provides structural benefits to the physical system of partial differential equations. Usually computing such closure of the system spends the majority of the total computation…
This work presents neural network based minimal entropy closures for the moment system of the Boltzmann equation, that preserve the inherent structure of the system of partial differential equations, such as entropy dissipation and hyperbolicity. The described method embeds convexity of the moment to entropy map in the neural network approximation to preserve the structure of the minimal entropy closure. Two techniques are used to implement the methods. The first approach approximates the map between moments and the minimal entropy of the moment system and is convex by design. The second appro…
We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally pro…
Extensions to the Navier-Stokes-Fourier Equations for Rarefied Transport: Variational Multiscale Moment Methods for the Boltzmann Equation
Oden Institute for Computational Sciences and Engineering, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算科学与工程研究所）；Department of Mechanical Engineering, Eindhoven University of Technology（埃因霍温理工大学机械工程系）
In a recent work, we proposed a hypothesis that the turbulence in gases could be produced by particles interacting via a potential - for example, the interatomic potential at short ranges, and the electrostatic potential at long ranges. Here, we examine the proposed mechanics of turbulence formation in a simple model of two particles, which interact solely via a potential. Following the kinetic theory approach, we derive a hierarchy of the velocity moment transport equations, and then truncate it via a novel closure based on the high Reynolds number condition. While standard closures of the ve…
