A smooth transition model between kinetic and hydrodynamic equations

This paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain. The idea is to use a buffer zone, in which both hydrodynamics and kinetic equations will be solved. The solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations. We use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone, thus no boundary condition is needed at the transition point. Consequently, this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations. A simple kinetic scheme is developed to discretize our model, and numerical examples are used to validate the method.

[1]  Axel Klar,et al.  Convergence of Alternating Domain Decomposition Schemes for Kinetic and Aerodynamic Equations , 1995 .

[2]  A. Klar.  Asymptotic analysis and coupling conditions for kinetic and hydrodynamic equations , 1998 .

[3]  L. Mieussens.  Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries , 2000 .

[4]  J. Schneider,et al.  Direct coupling of fluid and kinetic equations , 1996 .

[5]  Moulay D. Tidriri,et al.  Coupling Boltzmann and Navier-Stokes Equations by Friction , 1996 .

[6]  François Coron,et al.  Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation , 1989 .

[7]  François Golse,et al.  A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem , 2003 .

[8]  Axel Klar.  Domain Decomposition for Kinetic Problems with Nonequilibrium States , 1994 .

[9]  Benoît Perthame,et al.  Kinetic formulation of conservation laws , 2002 .

[10]  Youchun Qiu.  Etude des equations d'euler et de boltzmann et de leur couplage. Application a la simulation numerique d'ecoulements hypersoniques de gaz rarefies , 1993 .

[11]  A. Yamnahakki,et al.  SECOND ORDER BOUNDARY CONDITIONS FOR THE DRIFT-DIFFUSION EQUATIONS OF SEMICONDUCTORS , 1995 .

[12]  Patrick Le Tallec,et al.  Coupling Boltzmann and Navier-Stokes Equations by Half Fluxes , 1997 .

[13]  Shi Jin,et al.  A Smooth Transition Model between Kinetic and Diffusion Equations , 2004, SIAM J. Numer. Anal..

[14]  A. B. Huang,et al.  Test of statistical models for gases with and without internal energy states , 1973 .

[15]  Axel Klar,et al.  Transition from Kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms , 2000 .

[16]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[17]  H. C. Yee,et al.  A class of high resolution explicit and implicit shock-capturing methods , 1989 .

[18]  S. Tiwari,et al.  Coupling of the Boltzmann and Euler Equations with Automatic Domain Decomposition , 1998 .

[19]  M. Tidriri.  Rigorous Derivation and Analysis of Coupling of Kinetic Equations and Their Hydrodynamic Limits for a Simplified Boltzmann Model , 2001 .

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

[21]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

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