Convergence of a discrete-velocity model for the Boltzmann-BGK equation

Abstract We prove the convergence of a conservative and entropic discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. The main difficulty, due to its implicit definition, is to prove that this approximation is consistent. We also demonstrate the existence and uniqueness of a solution to the discrete-velocity model, by using a fixed-point theorem. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.

[1]  G. Sod.  Numerical methods in fluid dynamics , 1985 .

[2]  S. Mischler.  Uniqueness for the BGK-equation in $\mathbb{R}^N$ and rate of convergence for a semi-discrete scheme , 1996, Differential and Integral Equations.

[3]  M. Pulvirenti,et al.  Weighted L∞ bounds and uniqueness for the Boltzmann BGK model , 1993 .

[4]  Alexei Heintz,et al.  A new consistent discrete‐velocity model for the Boltzmann equation , 2002 .

[5]  C. Buet,et al.  A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics , 1996 .

[6]  D. Issautier,et al.  Convergence of a Weighted Particle Method for Solving the Boltzmann (B.G.K.) Equation , 1996 .

[7]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[8]  P. Raviart.  An analysis of particle methods , 1985 .

[9]  Stéphane Mischler.  Convergence of Discrete-Velocity Schemes for the Boltzmann Equation , 1997 .

[10]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[11]  Luc Mieussens,et al.  DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS , 2000 .

[12]  R. Gatignol.  Théorie cinétique des gaz à répartition discrète de vitesses , 1975 .

[13]  F. Rogier,et al.  A direct method for solving the Boltzmann equation , 1994 .

[14]  Andrzej Palczewski,et al.  A Consistency Result for a Discrete-Velocity Model of the Boltzmann Equation , 1997 .

[15]  P. Lions,et al.  On the Cauchy problem for Boltzmann equations: global existence and weak stability , 1989 .

[16]  Benoît Perthame,et al.  Global existence to the BGK model of Boltzmann equation , 1989 .

[17]  Juan-Chen Huang,et al.  Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations , 1995 .

[18]  Kazuo Aoki,et al.  Numerical Analysis of a Supersonic Rarefied Gas Flow past a Flat Plate at an Angle of Attack , 1996 .

[19]  P. Lions,et al.  Compactness in Boltzmann’s equation via Fourier integral operators and applications. III , 1994 .