A hybrid method for hydrodynamic-kinetic flow Part I: A particle-grid method for reducing stochastic noise in kinetic regimes

In this work we present a hybrid particle-grid Monte Carlo method for the Boltzmann equation, which is characterized by a significant reduction of the stochastic noise in the kinetic regime. The hybrid method is based on a first order splitting in time to separate the transport from the relaxation step. The transport step is solved by a deterministic scheme, while a hybrid DSMC-based method is used to solve the collision step. Such a hybrid scheme is based on splitting the solution in a collisional and a non-collisional part at the beginning of the collision step, and the DSMC method is used to solve the relaxation step for the collisional part of the solution only. This is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution, performing collisions as in a standard DSMC method, and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution. The latter is added to a piecewise constant reconstruction of the non-collisional part of the solution, which in fact remains unchanged during the relaxation step. Numerical results show that the stochastic noise is significantly reduced at large Knudsen numbers with respect to the standard DSMC method. Indeed in this algorithm, the particle scheme is applied only on the collisional part of the solution, so only this fraction of the solution is affected by stochastic fluctuations. But since the collisional part of the solution reduces as the Knudsen number increases, stochastic noise reduces as well at large Knudsen numbers.

[1]  F. G. Tcheremissine,et al.  Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation , 1998 .

[2]  B. Perthame,et al.  Introduction to the Theory of Random Particle Methods for Boltzmann Equation , 1996 .

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

[4]  Thomas E. Schwartzentruber,et al.  A modular particle-continuum numerical method for hypersonic non-equilibrium gas flows , 2007, J. Comput. Phys..

[5]  G. Russo,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2005 .

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

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

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

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

[10]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

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

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

[13]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : notes added in 1951 , 1951 .

[14]  H. Neunzert,et al.  On a simulation scheme for the Boltzmann equation , 1986 .

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

[16]  Giovanni Russo,et al.  Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation , 1997 .

[17]  Gary C. Cheng,et al.  Development and verification of a coupled DSMC-NS scheme using unstructured mesh , 2006, J. Comput. Phys..

[18]  Alejandro L. Garcia,et al.  Adaptive Mesh and Algorithm Refinement Using Direct Simulation Monte Carlo , 1999 .

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

[20]  G. Karniadakis,et al.  Microflows and Nanoflows: Fundamentals and Simulation , 2001 .

[21]  Axel Klar,et al.  An adaptive domain decomposition procedure for Boltzmann and Euler equations , 1998 .

[22]  Kenichi Nanbu,et al.  Direct simulation scheme derived from the Boltzmann equation. I - Monocomponent gases. II - Multicom , 1980 .

[23]  W. Steckelmacher.  Molecular gas dynamics and the direct simulation of gas flows , 1996 .

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

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

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

[27]  Benoît Perthame,et al.  An Introduction to Kinetic Schemes for Gas Dynamics , 1997, Theory and Numerics for Conservation Laws.

[28]  Lowell H. Holway,et al.  Kinetic Theory of Shock Structure Using an Ellipsoidal Distribution Function , 1965 .

[29]  Giacomo Dimarco,et al.  Hybrid multiscale methods for hyperbolic problems I. Hyperbolic relaxation problems , 2006 .

[30]  Patrick Le Tallec,et al.  Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases ☆ , 2002 .

[31]  Luc Mieussens,et al.  A smooth transition model between kinetic and hydrodynamic equations , 2005 .

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

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

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

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

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