# Moment Closure Approximations of the Boltzmann Equation Based on $$\varphi $$φ-Divergences

@article{Abdelmalik2015MomentCA, title={Moment Closure Approximations of the Boltzmann Equation Based on \$\$\varphi \$\$$\phi$-Divergences}, author={M. R. A. Abdelmalik and E. H. van Brummelen}, journal={Journal of Statistical Physics}, year={2015}, volume={164}, pages={77-104} }

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 $$\varphi $$φ-divergences and a corresponding closure procedure based on minimization of $$\varphi $$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…

## 16 Citations

### An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation

- Computer ScienceComput. Math. Appl.
- 2016

### A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis

- MathematicsKinetic and Related Models
- 2022

A closure relation for moments equations in kinetic theory was recently introduced in [38], based on the study of the geometry of the set of moments. This relation was constructed from a projection…

### Entropy stable Hermite approximation of the linearised Boltzmann equation for inflow and outflow boundaries

- Computer ScienceJ. Comput. Phys.
- 2018

### An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics

- MathematicsInternational Journal of Computational Fluid Dynamics
- 2021

The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment…

### Entropy Stable Discontinuous Galerkin Finite Element Moment Methods for Compressible Fluid Dynamics

- Computer Science, Physics
- 2020

This work proposes numerical approximations of the Boltzmann equation that are consistent with the Euler and Navier–Stoke–Fourier solutions and presents a numerical approximation that is based on the discontinuous Galerkin method in position dependence and on the renormalized-moment method in velocity dependence.

### Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region

- MathematicsContinuum Mechanics and Thermodynamics
- 2019

The rational extended thermodynamics theory describes non-equilibrium phenomena for rarefied gases, and it is usually approximated in the neighborhood of an equilibrium state. Consequently, the…

### The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

- MathematicsJournal of Nonlinear Mathematical Physics
- 2022

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell…

### Thermodynamically consistent coarse-graining of polar active fluids

- PhysicsPhysical Review Fluids
- 2022

We introduce a closure model for coarse-grained kinetic theories of polar active ﬂuids. Based on a quasi-equilibrium approximation of the particle distribution function, the model closely captures…

### Investigation of Aerodynamic Characteristics of Aircrafts in a Rarefied Gas Flow Using the Moment Method

- MathematicsInt. J. Math. Math. Sci.
- 2022

A one-dimensional nonstationary nonlinear moment system of equations and an approximation of Maxwell’s microscopic boundary condition will be introduced. The flight speed and surface temperature of…

## References

SHOWING 1-10 OF 64 REFERENCES

### An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation

- Computer ScienceComput. Math. Appl.
- 2016

### Affordable robust moment closures for CFD based on the maximum-entropy hierarchy

- MathematicsJ. Comput. Phys.
- 2013

### Moment closure hierarchies for kinetic theories

- Mathematics
- 1996

This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the…

### Scale-Induced Closure for Approximations of Kinetic Equations

- Mathematics
- 2010

The order-of-magnitude method proposed by Struchtrup (Phys. Fluids 16(11):3921–3934, 2004) is a new closure procedure for the infinite moment hierarchy in kinetic theory of gases, taking into account…

### Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-variate Pearson-IV-distributions

- Mathematics
- 2009

In this paper we develop a new closure theory for moment approximations in kinetic gas theory and derive hyperbolic moment equations for 13 fluid variables including stress and heat flux. Classical…

### Fluid dynamic limits of kinetic equations. I. Formal derivations

- Mathematics, Physics
- 1991

The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a…

### Entropic approximation in kinetic theory

- Mathematics
- 2004

Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the…

### Entropy principle for the moment systems of degree $\alpha$ associated to the Boltzmann equation. Critical derivatives and non controllable boundary data

- Mathematics
- 2002

m moments and of degree
$\alpha (ET_m^\alpha)$. For such theories the entropy principle is still valid only, if the non equilibrium field variables and their derivatives are sufficiently small with…

### Modeling Nonequilibrium Gas Flow Based on Moment Equations

- Physics
- 2016

This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present…

### Regularization of Grad’s 13 moment equations: Derivation and linear analysis

- Physics
- 2003

A new closure for Grad’s 13 moment equations is presented that adds terms of Super-Burnett order to the balances of pressure deviator and heat flux vector. The additional terms are derived from…