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June  2013, 6(2): 291-315. doi: 10.3934/krm.2013.6.291

The moment guided Monte Carlo method for the Boltzmann equation

Giacomo Dimarco 1,

1. 

Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, 118, route de Narbonne 31062 TOULOUSE Cedex, France

Received  July 2012 Revised  October 2012 Published  February 2013

In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.
Keywords: Monte Carlo methods, hybrid methods, variance reduction, Boltzmann equation, moments closure, fluid equations..
Mathematics Subject Classification: Primary: 76P05, 65C20, 65C0.
Citation: Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
References:
[1]

H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233. doi: 10.1002/mma.1670080114.

[2]

G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows," Oxford Engineering Science Series, 42, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[3]

C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation," Institute of Physics (IOP), Series in Plasma Physics, 2004.

[4]

J.-F. Bourgat, P. Le Tallec and M. D. Tidriri, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127 (1996), 227-245. doi: 10.1006/jcph.1996.0172.

[5]

J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475.

[6]

R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods, in "Acta Numerica, 1998," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49. doi: 10.1017/S0962492900002804.

[7]

R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics, in "Transport in Transition Regimes" (Minneapolis, MN, 2000), IMA Vol. App. Math., 135, Springer, New York, (2004), 57-73. doi: 10.1007/978-1-4613-0017-5_3.

[8]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[9]

A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, to appear in KRM, 2012.

[10]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the gas-dynamics Boltzmann BGK equation, J. Comput. Phys., 199 (2004), 776-808. doi: 10.1016/j.jcp.2004.03.007.

[11]

P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method, Int. J. Num. Meth. Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345.

[12]

P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Model. Simul., 5 (2006), 940-979. doi: 10.1137/060651574.

[13]

P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo, J. Comp. Phys., 231 (2012), 2414-2437. doi: 10.1016/j.jcp.2011.11.030.

[14]

P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comp. Phys., 229 (2010), 4907-4933. doi: 10.1016/j.jcp.2010.03.009.

[15]

G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems, Comm. Math. Sci., 1 (2006), 155-177.

[16]

G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634. doi: 10.1137/080730585.

[17]

G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J. Num. Anal., 49 (2011), 2057-2077. doi: 10.1137/100811052.

[18]

F. Filbet and S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017.

[19]

F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x.

[20]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132.

[21]

D. B. Hash and H. A. Hassan, Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods, J. Thermophys. Heat Transf., 10 (1996), 242-249.

[22]

T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comp. Phys., 226 (2007), 2341-2358. doi: 10.1016/j.jcp.2007.07.006.

[23]

T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701.

[24]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[25]

S. Jin and Z. P. Xin, Relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276. doi: 10.1002/cpa.3160480303.

[26]

P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67. doi: 10.1006/jcph.1997.5729.

[27]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics, Birkhäuser Verlag, Basel, 1990.

[28]

S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer, 2004.

[29]

K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049.

[30]

D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), 231-244.

[31]

S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition, J. Comput. Phys., 144 (1998), 710-726. doi: 10.1006/jcph.1998.6011.

[32]

S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations, J. Comput. Phys., 228 (2009), 7109-7124. doi: 10.1016/j.jcp.2009.06.019.

show all references

References:
[1]

H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233. doi: 10.1002/mma.1670080114.

[2]

G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows," Oxford Engineering Science Series, 42, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[3]

C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation," Institute of Physics (IOP), Series in Plasma Physics, 2004.

[4]

J.-F. Bourgat, P. Le Tallec and M. D. Tidriri, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127 (1996), 227-245. doi: 10.1006/jcph.1996.0172.

[5]

J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475.

[6]

R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods, in "Acta Numerica, 1998," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49. doi: 10.1017/S0962492900002804.

[7]

R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics, in "Transport in Transition Regimes" (Minneapolis, MN, 2000), IMA Vol. App. Math., 135, Springer, New York, (2004), 57-73. doi: 10.1007/978-1-4613-0017-5_3.

[8]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[9]

A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, to appear in KRM, 2012.

[10]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the gas-dynamics Boltzmann BGK equation, J. Comput. Phys., 199 (2004), 776-808. doi: 10.1016/j.jcp.2004.03.007.

[11]

P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method, Int. J. Num. Meth. Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345.

[12]

P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Model. Simul., 5 (2006), 940-979. doi: 10.1137/060651574.

[13]

P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo, J. Comp. Phys., 231 (2012), 2414-2437. doi: 10.1016/j.jcp.2011.11.030.

[14]

P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comp. Phys., 229 (2010), 4907-4933. doi: 10.1016/j.jcp.2010.03.009.

[15]

G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems, Comm. Math. Sci., 1 (2006), 155-177.

[16]

G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634. doi: 10.1137/080730585.

[17]

G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J. Num. Anal., 49 (2011), 2057-2077. doi: 10.1137/100811052.

[18]

F. Filbet and S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017.

[19]

F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x.

[20]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132.

[21]

D. B. Hash and H. A. Hassan, Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods, J. Thermophys. Heat Transf., 10 (1996), 242-249.

[22]

T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comp. Phys., 226 (2007), 2341-2358. doi: 10.1016/j.jcp.2007.07.006.

[23]

T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701.

[24]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[25]

S. Jin and Z. P. Xin, Relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276. doi: 10.1002/cpa.3160480303.

[26]

P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67. doi: 10.1006/jcph.1997.5729.

[27]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics, Birkhäuser Verlag, Basel, 1990.

[28]

S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer, 2004.

[29]

K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049.

[30]

D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), 231-244.

[31]

S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition, J. Comput. Phys., 144 (1998), 710-726. doi: 10.1006/jcph.1998.6011.

[32]

S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations, J. Comput. Phys., 228 (2009), 7109-7124. doi: 10.1016/j.jcp.2009.06.019.

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