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A discrete kinetic approximation for the incompressible Navier-Stokes equations

Maria Francesca Carfora, Roberto Natalini (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

A geometric lower bound on Grad’s number

Alessio Figalli (2009)

ESAIM: Control, Optimisation and Calculus of Variations

In this note we provide a new geometric lower bound on the so-called Grad’s number of a domain Ø in terms of how far Ø is from being axisymmetric. Such an estimate is important in the study of the trend to equilibrium for the Boltzmann equation for dilute gases.

A geometric lower bound on Grad's number

Alessio Figalli (2008)

ESAIM: Control, Optimisation and Calculus of Variations

In this note we provide a new geometric lower bound on the so-called Grad's number of a domain Ω in terms of how far Ω is from being axisymmetric. Such an estimate is important in the study of the trend to equilibrium for the Boltzmann equation for dilute gases.

A Langevin Description for Driven Granular Gases

P. Maynar, M. I. García de Soria (2011)

Mathematical Modelling of Natural Phenomena

The study of the fluctuations in the steady state of a heated granular system is reviewed. A Boltzmann-Langevin description can be built requiring consistency with the equations for the one- and two-particle correlation functions. From the Boltzmann-Langevin equation, Langevin equations for the total energy and the transverse velocity field are derived. The existence of a fluctuation-dissipation relation for the transverse velocity field is also...

Around 3D Boltzmann non linear operator without angular cutoff, a new formulation

Radjesvarane Alexandre (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs, with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.

Characterization of collision kernels

Laurent Desvillettes, Francesco Salvarani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we show how abstract physical requirements are enough to characterize the classical collision kernels appearing in kinetic equations. In particular Boltzmann and Landau kernels are derived.

Characterization of collision kernels

Laurent Desvillettes, Francesco Salvarani (2003)

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

In this paper we show how abstract physical requirements are enough to characterize the classical collision kernels appearing in kinetic equations. In particular Boltzmann and Landau kernels are derived.

Conservative forms of Boltzmann's collision operator: Landau revisited

Cédric Villani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We show that Boltzmann's collision operator can be written explicitly in divergence and double divergence forms. These conservative formulations may be of interest for both theoretical and numerical purposes. We give an application to the asymptotics of grazing collisions.

Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation

Clément Mouhot, Lorenzo Pareschi, Thomas Rey (2013)

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

Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N2d + 1) where d is the dimension of the velocity space. In this paper, following the ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math. Comput. 75 (2006) 1833–1852], we derive fast summation techniques for the evaluation of...

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