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On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger, Sebastian Noelle, Susanne Zimmermann (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation...

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger, Sebastian Noelle, Susanne Zimmermann (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the...

On the motion of a body in thermal equilibrium immersed in a perfect gas

Kazuo Aoki, Guido Cavallaro, Carlo Marchioro, Mario Pulvirenti (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity V ∞ and prove that, under suitable smallness assumptions, the approach...

On the one-dimensional Boltzmann equation for granular flows

Dario Benedetto, Mario Pulvirenti (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.

On the stationary Boltzmann equation

Leif Arkeryd (2001/2002)

Séminaire Équations aux dérivées partielles

For stationary kinetic equations, entropy dissipation can sometimes be used in existence proofs similarly to entropy in the time dependent situation. Recent results in this spirit obtained in collaboration with A. Nouri, are here presented for the nonlinear stationary Boltzmann equation in bounded domains of I R n with given indata and diffuse reflection on the boundary.

One-dimensional kinetic models of granular flows

Giuseppe Toscani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance...

Recent results on the Boltzmann equation

Carlo Cercignani (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the last few years the theory of the nonlinear Boltzmann equation has witnessed a veritable turrent of contributions, spurred by the basic result of DiPerna and Lions. Here we wish to survey these results with particular attention to some recent developments.

Role of Molecular Chaos in Granular Fluctuating Hydrodynamics

G. Costantini, A. Puglisi (2011)

Mathematical Modelling of Natural Phenomena

We perform a numerical study of the fluctuations of the rescaled hydrodynamic transverse velocity field during the cooling state of a homogeneous granular gas. We are interested in the role of Molecular Chaos for the amplitude of the hydrodynamic noise and its relaxation in time. For this purpose we compare the results of Molecular Dynamics (MD, deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, random process), where Molecular...

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