On Deterministic Approximation of the Boltzmann Equation in a Bounded Domain

In this paper we present a fully deterministic method for the numerical solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain for multi-scale problems. Periodic, specular reflection and diffusive boundary conditions are discussed and investigated numerically. The collision operator is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity with a computational cost of $N\,\log(N)$, where $N$ is the number of degree of freedom in velocity space. This algorithm is coupled with a second order finite volume scheme in space and a time discretization allowing to deal for rarefied regimes as well as their hydrodynamic limit. Finally, several numerical tests illustrate the efficiency and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects, trend to equilibrium).

[1]  Sergej Rjasanow,et al.  Numerical solution of the Boltzmann equation on the uniform grid , 2002, Computing.

[2]  G. Toscani,et al.  Fast spectral methods for the Fokker-Planck-Landau collision operator , 2000 .

[3]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[4]  Lorenzo Pareschi,et al.  Solving the Boltzmann Equation in N log2N , 2006, SIAM J. Sci. Comput..

[5]  T. Carleman,et al.  Sur la théorie de l'équation intégrodifférentielle de Boltzmann , 1933 .

[6]  A. Bobylev.  Quasistationary hydrodynamics for the Boltzmann equation , 1995 .

[7]  Lorenzo Pareschi,et al.  A Fourier spectral method for homogeneous boltzmann equations , 1996 .

[8]  L. Pareschi,et al.  On the stability of spectral methods for the homogeneous Boltzmann equation , 2000 .

[9]  Irene M. Gamba,et al.  SHOCK AND BOUNDARY STRUCTURE FORMATION BY SPECTRAL-LAGRANGIAN METHODS FOR THE INHOMOGENEOUS BOLTZMANN TRANSPORT EQUATION * , 2010 .

[10]  Lorenzo Pareschi,et al.  Fast algorithms for computing the Boltzmann collision operator , 2006, Math. Comput..

[11]  Francis Filbet,et al.  High order numerical methods for the space non-homogeneous Boltzmann equation , 2003 .

[12]  Lorenzo Pareschi,et al.  Accurate numerical methods for the collisional motion of (heated) granular flows , 2005 .

[13]  S. Rjasanow,et al.  Fast deterministic method of solving the Boltzmann equation for hard spheres , 1999 .

[14]  Kenichi Nanbu,et al.  Direct Simulation Scheme Derived from the Boltzmann Equation. II. Multicomponent Gas Mixtures , 1980 .

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

[16]  Francis Filbet,et al.  Accurate numerical methods for the Boltzmann equation , 2004 .

[17]  Shi Jin,et al.  A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources , 2009, J. Comput. Phys..

[18]  Birkhauser.  Modeling and Computational Methods for Kinetic Equations , 2004 .

[19]  Yan Guo,et al.  Almost Exponential Decay Near Maxwellian , 2006 .

[20]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[21]  Lorenzo Pareschi,et al.  Modeling and Computational Methods for Kinetic Equations , 2012 .

[22]  L. C. Pitchford,et al.  A Numerical Solution of the Boltzmann Equation , 1983 .

[23]  Lorenzo Pareschi,et al.  Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit , 2003 .

[24]  Lorenzo Pareschi,et al.  A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case , 2002 .

[25]  Cédric Villani,et al.  On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation , 2005 .

[26]  S. Rjasanow,et al.  Difference scheme for the Boltzmann equation based on the Fast Fourier Transform , 1997 .

[27]  Michail A. Gallis,et al.  Direct simulation Monte Carlo convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow , 2006 .

[28]  P. Bertrand,et al.  Conservative numerical schemes for the Vlasov equation , 2001 .

[29]  Giacomo Dimarco,et al.  Exponential Runge-Kutta Methods for Stiff Kinetic Equations , 2010, SIAM J. Numer. Anal..

[30]  F. Uribe,et al.  Burnett description for plane Poiseuille flow. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  S. Takata,et al.  Inappropriateness of the heat‐conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation , 1996 .

[32]  Irene M. Gamba,et al.  Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states , 2009, J. Comput. Phys..

[33]  Yoshio Sone,et al.  Molecular gas dynamics , 2007 .

[34]  Katsuhisa Koura,et al.  Comment on “Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases” , 1981 .

[35]  Lorenzo Pareschi,et al.  Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator , 2000, SIAM J. Numer. Anal..