The moment guided Monte Carlo method for the Boltzmann equation

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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

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

Abstract
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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 & 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. doi: 10.1002/mma.1670080114. Google Scholar
[2]	G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows,", Oxford Engineering Science Series, 42 (1995). Google Scholar
[3]	C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", Institute of Physics (IOP), (2004). Google Scholar
[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. doi: 10.1006/jcph.1996.0172. Google Scholar
[5]	J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation,, J. Comput. Phys., 228 (2009), 460. Google Scholar
[6]	R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods,, in, 7 (1998), 1. doi: 10.1017/S0962492900002804. Google Scholar
[7]	R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics,, in, 135 (2004), 57. doi: 10.1007/978-1-4613-0017-5_3. Google Scholar
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar
[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). Google Scholar
[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. doi: 10.1016/j.jcp.2004.03.007. Google Scholar
[11]	P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method,, Int. J. Num. Meth. Fluids, 67 (2011), 189. doi: 10.1002/fld.2345. Google Scholar
[12]	P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects,, Multiscale Model. Simul., 5 (2006), 940. doi: 10.1137/060651574. Google Scholar
[13]	P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo,, J. Comp. Phys., 231 (2012), 2414. doi: 10.1016/j.jcp.2011.11.030. Google Scholar
[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. doi: 10.1016/j.jcp.2010.03.009. Google Scholar
[15]	G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems,, Comm. Math. Sci., 1 (2006), 155. Google Scholar
[16]	G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations,, SIAM J. Sci. Comput., 32 (2010), 603. doi: 10.1137/080730585. Google Scholar
[17]	G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations,, SIAM J. Num. Anal., 49 (2011), 2057. doi: 10.1137/100811052. Google Scholar
[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. doi: 10.1016/j.jcp.2010.06.017. Google Scholar
[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. doi: 10.1007/s10915-010-9394-x. Google Scholar
[20]	W. E and B. Engquist, The heterogeneous multiscale methods,, Comm. Math. Sci., 1 (2003), 87. Google Scholar
[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. Google Scholar
[22]	T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation,, J. Comp. Phys., 226 (2007), 2341. doi: 10.1016/j.jcp.2007.07.006. Google Scholar
[23]	T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo,, Phys. Fluids, 19 (2007). Google Scholar
[24]	S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. doi: 10.1137/S1064827598334599. Google Scholar
[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. doi: 10.1002/cpa.3160480303. Google Scholar
[26]	P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes,, J. Comput. Phys., 136 (1997), 51. doi: 10.1006/jcph.1997.5729. Google Scholar
[27]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Lectures in Mathematics, (1990). Google Scholar
[28]	S. Liu, "Monte Carlo Strategies in Scientific Computing,", Springer, (2004). Google Scholar
[29]	K. Nanbu, Direct simulation scheme derived from the Boltzmann equation,, J. Phys. Soc. Japan, 49 (1980), 2042. Google Scholar
[30]	D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow,, J. Comput. Phys., 34 (1980), 231. Google Scholar
[31]	S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition,, J. Comput. Phys., 144 (1998), 710. doi: 10.1006/jcph.1998.6011. Google Scholar
[32]	S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations,, J. Comput. Phys., 228 (2009), 7109. doi: 10.1016/j.jcp.2009.06.019. Google Scholar

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References:

[1]	H. Babovsky, On a simulation scheme for the Boltzmann equation,, Math. Methods Appl. Sci., 8 (1986), 223. doi: 10.1002/mma.1670080114. Google Scholar
[2]	G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows,", Oxford Engineering Science Series, 42 (1995). Google Scholar
[3]	C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", Institute of Physics (IOP), (2004). Google Scholar
[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. doi: 10.1006/jcph.1996.0172. Google Scholar
[5]	J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation,, J. Comput. Phys., 228 (2009), 460. Google Scholar
[6]	R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods,, in, 7 (1998), 1. doi: 10.1017/S0962492900002804. Google Scholar
[7]	R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics,, in, 135 (2004), 57. doi: 10.1007/978-1-4613-0017-5_3. Google Scholar
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). doi: 10.1007/978-1-4612-1039-9. Google Scholar
[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). Google Scholar
[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. doi: 10.1016/j.jcp.2004.03.007. Google Scholar
[11]	P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method,, Int. J. Num. Meth. Fluids, 67 (2011), 189. doi: 10.1002/fld.2345. Google Scholar
[12]	P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects,, Multiscale Model. Simul., 5 (2006), 940. doi: 10.1137/060651574. Google Scholar
[13]	P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo,, J. Comp. Phys., 231 (2012), 2414. doi: 10.1016/j.jcp.2011.11.030. Google Scholar
[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. doi: 10.1016/j.jcp.2010.03.009. Google Scholar
[15]	G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems,, Comm. Math. Sci., 1 (2006), 155. Google Scholar
[16]	G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations,, SIAM J. Sci. Comput., 32 (2010), 603. doi: 10.1137/080730585. Google Scholar
[17]	G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations,, SIAM J. Num. Anal., 49 (2011), 2057. doi: 10.1137/100811052. Google Scholar
[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. doi: 10.1016/j.jcp.2010.06.017. Google Scholar
[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. doi: 10.1007/s10915-010-9394-x. Google Scholar
[20]	W. E and B. Engquist, The heterogeneous multiscale methods,, Comm. Math. Sci., 1 (2003), 87. Google Scholar
[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. Google Scholar
[22]	T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation,, J. Comp. Phys., 226 (2007), 2341. doi: 10.1016/j.jcp.2007.07.006. Google Scholar
[23]	T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo,, Phys. Fluids, 19 (2007). Google Scholar
[24]	S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. doi: 10.1137/S1064827598334599. Google Scholar
[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. doi: 10.1002/cpa.3160480303. Google Scholar
[26]	P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes,, J. Comput. Phys., 136 (1997), 51. doi: 10.1006/jcph.1997.5729. Google Scholar
[27]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Lectures in Mathematics, (1990). Google Scholar
[28]	S. Liu, "Monte Carlo Strategies in Scientific Computing,", Springer, (2004). Google Scholar
[29]	K. Nanbu, Direct simulation scheme derived from the Boltzmann equation,, J. Phys. Soc. Japan, 49 (1980), 2042. Google Scholar
[30]	D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow,, J. Comput. Phys., 34 (1980), 231. Google Scholar
[31]	S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition,, J. Comput. Phys., 144 (1998), 710. doi: 10.1006/jcph.1998.6011. Google Scholar
[32]	S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations,, J. Comput. Phys., 228 (2009), 7109. doi: 10.1016/j.jcp.2009.06.019. Google Scholar

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