Fluid Solver Independent Hybrid Methods for Multiscale Kinetic Equations

In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I. Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G. Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems. Here, at variance with respect to the previous methods, we construct a method form-fitting to any type of finite volume or finite difference scheme for the reduced equilibrium system. Thanks to the coupling of Monte Carlo techniques for the solution of the kinetic equations with macroscopic methods for the limiting fluid equations, we show how it is possible to solve multiscale fluid dynamic phenomena faster with respect to traditional deterministic/stochastic methods for the full kinetic equations. In addition, due to the hybrid nature of the schemes, the numerical solution is affected by less fluctuations when compared to standard Monte Carlo schemes. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods in comparison with classical approaches used in the simulation of kinetic equations.

[1]  L. Pareschi,et al.  HYBRID MULTISCALE METHODS I. HYPERBOLIC RELAXATION PROBLEMS∗ , 2006 .

[2]  B. Perthame Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions , 1992 .

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

[4]  David B. Goldstein,et al.  Hybrid Euler/Direct Simulation Monte Carlo Calculation of Unsteady Slit Flow , 2000 .

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

[6]  P.,et al.  A hybrid kinetic-fluid model for solving the gas dynamics Boltzmann-BGK equation , 2004 .

[7]  B. Perthame,et al.  Boltzmann type schemes for gas dynamics and the entropy property , 1990 .

[8]  D. Pullin,et al.  Generation of normal variates with given sample mean and variance , 1979 .

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

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

[11]  Gabriella Puppo,et al.  Implicit–Explicit Schemes for BGK Kinetic Equations , 2007, J. Sci. Comput..

[12]  C. Cercignani Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations , 2000 .

[13]  Giacomo Dimarco,et al.  Hybrid Multiscale Methods II. Kinetic Equations , 2008, Multiscale Model. Simul..

[14]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[15]  Giacomo Dimarco,et al.  Domain Decomposition Techniques and Hybrid Multiscale Methods for Kinetic Equations , 2008 .

[16]  Lorenzo Pareschi,et al.  Time Relaxed Monte Carlo Methods for the Boltzmann Equation , 2001, SIAM J. Sci. Comput..

[17]  Cédric Villani Rarefied Gas Dynamics: From basic concepts to actual calculations: (Cambridge texts in applied mathematics, Cambridge University Press, 2000, 320 pp.) £ 18.95; US$ 29.95 paperback ISBN 0 521 65992 2 , 2001 .

[18]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[19]  Lorenzo Pareschi,et al.  Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods , 2005 .

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

[21]  G. Bird Molecular Gas Dynamics and the Direct Simulation of Gas Flows , 1994 .

[22]  Patrick Le Tallec,et al.  Coupling Boltzmann and Euler equations without overlapping , 1992 .

[23]  S. M. Deshpande,et al.  A second-order accurate kinetic-theory-based method for inviscid compressible flows , 1986 .

[24]  Shi Jin Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1995 .

[25]  Luc Mieussens,et al.  A moving interface method for dynamic kinetic-fluid coupling , 2007, J. Comput. Phys..

[26]  B. Perthame,et al.  Numerical passage from kinetic to fluid equations , 1991 .

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

[28]  R. Caflisch Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[29]  Lorenzo Pareschi,et al.  Hybrid Multiscale Methods for Hyperbolic and Kinetic Problems , 2005 .