The moment guided Monte Carlo method for the Boltzmann equation

Previous Article
Nonlinear stability of a Vlasov equation for magnetic plasmas
KRM Home
This Issue
Next Article
Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups

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
Related Papers

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.
[2]	G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows,", Oxford Engineering Science Series, 42 (1995).
[3]	C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", Institute of Physics (IOP), (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. 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.
[6]	R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods,, in, 7 (1998), 1. doi: 10.1017/S0962492900002804.
[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.
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (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. 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. 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. 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. 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. 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.
[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.
[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.
[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.
[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.
[20]	W. E and B. Engquist, The heterogeneous multiscale methods,, Comm. Math. Sci., 1 (2003), 87.
[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.
[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.
[23]	T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo,, Phys. Fluids, 19 (2007).
[24]	S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. 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. 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. doi: 10.1006/jcph.1997.5729.
[27]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Lectures in Mathematics, (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.
[30]	D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow,, J. Comput. Phys., 34 (1980), 231.
[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.
[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.

show all references

References:

[1]	H. Babovsky, On a simulation scheme for the Boltzmann equation,, Math. Methods Appl. Sci., 8 (1986), 223. doi: 10.1002/mma.1670080114.
[2]	G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows,", Oxford Engineering Science Series, 42 (1995).
[3]	C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", Institute of Physics (IOP), (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. 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.
[6]	R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods,, in, 7 (1998), 1. doi: 10.1017/S0962492900002804.
[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.
[8]	C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (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. 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. 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. 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. 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. 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.
[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.
[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.
[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.
[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.
[20]	W. E and B. Engquist, The heterogeneous multiscale methods,, Comm. Math. Sci., 1 (2003), 87.
[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.
[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.
[23]	T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo,, Phys. Fluids, 19 (2007).
[24]	S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. 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. 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. doi: 10.1006/jcph.1997.5729.
[27]	R. J. LeVeque, "Numerical Methods for Conservation Laws,", Lectures in Mathematics, (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.
[30]	D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow,, J. Comput. Phys., 34 (1980), 231.
[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.
[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.

[1]	Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204
[2]	M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013
[3]	Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
[4]	Milana Pavić-Čolić, Maja Tasković. Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules. Kinetic & Related Models, 2018, 11 (3) : 597-613. doi: 10.3934/krm.2018025
[5]	Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055
[6]	Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[7]	Jérôme Droniou. Remarks on discretizations of convection terms in Hybrid mimetic mixed methods. Networks & Heterogeneous Media, 2010, 5 (3) : 545-563. doi: 10.3934/nhm.2010.5.545
[8]	Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[9]	Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-23. doi: 10.3934/dcdsb.2018335
[10]	B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463
[11]	Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[12]	Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[13]	Lijian Jiang, Yalchin Efendiev, Victor Ginting. Multiscale methods for parabolic equations with continuum spatial scales. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 833-859. doi: 10.3934/dcdsb.2007.8.833
[14]	Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[15]	Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839
[16]	Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289
[17]	Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125
[18]	Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[19]	Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803
[20]	Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025

2017 Impact Factor: 1.219

American Institute of Mathematical Sciences