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December  2012, 5(4): 787-816. doi: 10.3934/krm.2012.5.787

Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

Anaïs Crestetto 1, , Nicolas Crouseilles 2, and  Mohammed Lemou 3,

1. 

INRIA-Nancy Grand Est, CALVI Project, and IRMA, Université de Strasbourg, 67084, STRASBOURG, France
2. 

INRIA-Rennes Bretagne Atlantique, IPSO Project, and IRMAR (Université de Rennes 1), 35042 RENNES, France
3. 

CNRS and IRMAR (Université de Rennes 1), and INRIA-Rennes Bretagne Atlantique, IPSO Project, 35042 RENNES, France

Received  April 2012 Revised  June 2012 Published  November 2012

This work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.
Keywords: particles method., micro-macro decomposition, BGK equation, Euler equation, asymptotic preserving schemes, Vlasov equation, Chapman-Enskog expansion.
Mathematics Subject Classification: 65M06, 35B25, 82C80, 82D10, 41A6.
Citation: Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic and Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787
References:
[1]

C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I. Formal derivation, J. Statist. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608.

[2]

C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation, Institute of Physics (IOP), Series in Plasma Physics 2004.

[3]

M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032.

[4]

S. Brunner, E. Valeo and J. A. Krommes, Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. of Plasmas, 12 (1999).

[5]

S. Brunner, E. Valeo and J. A. Krommes, Linear delta-f simulations of nonlocal electron heat transport, Phys. of Plasmas, 7 (2000).

[6]

J.-M. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26-42.

[7]

G.-H. Cottet and P.-A. Raviart, On particle-in-cell methods for the one-dimensional Vlasov-Poisson equations, SIAM J. of Numer. Anal., 21 (1984), 52-76.

[8]

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.

[9]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: Diffusion and high-field scaling limits, Kin. Rel. Models, 4 (2011), 441-477.

[10]

N. Crouseilles, M. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for the Vlasov equation, J. Comput. Phys., 229 (2010), 1927-1953. doi: 10.1016/j.jcp.2009.11.007.

[11]

P. Degond and F. J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. on Scientific and Statistical Computing, 11 (1990), 293-310. doi: 10.1137/0911018.

[12]

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

[13]

P. Degond, G. Dimarco and L. Pareschi, The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345.

[14]

P. Degond, S. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), 665-694. doi: 10.1016/j.jcp.2005.03.025.

[15]

P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations, Part 2: The anisotropic case, Math. Comput., 53 (1989), 509-525. doi: 10.1090/S0025-5718-1989-0983560-5.

[16]

F. Delaurens and F. J. Mustieles, A deterministic particle method for solving kinetic transport equations: The semiconductor Boltzmann equation case, SIAM J. Appl. Math., 52 (1991), 973-988. doi: 10.1137/0152056.

[17]

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.

[18]

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

[19]

E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal., 34 (1997), 2168-2194. doi: 10.1137/S0036142995287768.

[20]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1994), 2099-2119. doi: 10.1137/S0036142994266856.

[21]

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

[22]

M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Math\'ematique, 348 (2010), 455-460.

[23]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comp., 31 (2008), 334-368. doi: 10.1137/07069479X.

[24]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics Birkhauser Verlag, Basel, 1992.

[25]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models and Meth. Appl. Sci., 8 (2000), 1121-1149.

[26]

B. Niclot, P. Degond and F. Poupaud, Deterministic particle simulations of the Boltzmann transport equation of semiconductors, J. Comput. Phys., 78 (1988), 313-349. doi: 10.1016/0021-9991(88)90053-8.

[27]

L. Pareschi and G. Russo, Asymptotic preserving Monte Carlo methods for the Boltzmann equation, Trans. Theo. Stat. Phys., 29 (2000), 415-430. doi: 10.1080/00411450008205882.

[28]

L. Pareschi and G. Russo, Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput., 23 (2001), 1253-1273. doi: 10.1137/S1064827500375916.

[29]

P.-A. Raviart, "An Analysis of Particle Methods in Numerical Methods in Fluid Dynamics," Lecture Notes in Math. 1127, F. Brezzi ed., Springer-Verlag, Berlin, 1985.

[30]

E. Sonnendrücker, Mathematical models for fusion,, Lecture Notes, (). 

[31]

B. Yan and S. Jin, A successive penalty-based asymptotic-preserving scheme for kinetic equations,, submitted., (). 

show all references

References:
[1]

C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I. Formal derivation, J. Statist. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608.

[2]

C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation, Institute of Physics (IOP), Series in Plasma Physics 2004.

[3]

M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032.

[4]

S. Brunner, E. Valeo and J. A. Krommes, Collisional delta-f scheme with evolving background for transport time scale simulations, Phys. of Plasmas, 12 (1999).

[5]

S. Brunner, E. Valeo and J. A. Krommes, Linear delta-f simulations of nonlocal electron heat transport, Phys. of Plasmas, 7 (2000).

[6]

J.-M. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26-42.

[7]

G.-H. Cottet and P.-A. Raviart, On particle-in-cell methods for the one-dimensional Vlasov-Poisson equations, SIAM J. of Numer. Anal., 21 (1984), 52-76.

[8]

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.

[9]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: Diffusion and high-field scaling limits, Kin. Rel. Models, 4 (2011), 441-477.

[10]

N. Crouseilles, M. Mehrenberger and E. Sonnendrücker, Conservative semi-Lagrangian schemes for the Vlasov equation, J. Comput. Phys., 229 (2010), 1927-1953. doi: 10.1016/j.jcp.2009.11.007.

[11]

P. Degond and F. J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. on Scientific and Statistical Computing, 11 (1990), 293-310. doi: 10.1137/0911018.

[12]

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

[13]

P. Degond, G. Dimarco and L. Pareschi, The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345.

[14]

P. Degond, S. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), 665-694. doi: 10.1016/j.jcp.2005.03.025.

[15]

P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations, Part 2: The anisotropic case, Math. Comput., 53 (1989), 509-525. doi: 10.1090/S0025-5718-1989-0983560-5.

[16]

F. Delaurens and F. J. Mustieles, A deterministic particle method for solving kinetic transport equations: The semiconductor Boltzmann equation case, SIAM J. Appl. Math., 52 (1991), 973-988. doi: 10.1137/0152056.

[17]

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.

[18]

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

[19]

E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal., 34 (1997), 2168-2194. doi: 10.1137/S0036142995287768.

[20]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1994), 2099-2119. doi: 10.1137/S0036142994266856.

[21]

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

[22]

M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Math\'ematique, 348 (2010), 455-460.

[23]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comp., 31 (2008), 334-368. doi: 10.1137/07069479X.

[24]

R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics Birkhauser Verlag, Basel, 1992.

[25]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models and Meth. Appl. Sci., 8 (2000), 1121-1149.

[26]

B. Niclot, P. Degond and F. Poupaud, Deterministic particle simulations of the Boltzmann transport equation of semiconductors, J. Comput. Phys., 78 (1988), 313-349. doi: 10.1016/0021-9991(88)90053-8.

[27]

L. Pareschi and G. Russo, Asymptotic preserving Monte Carlo methods for the Boltzmann equation, Trans. Theo. Stat. Phys., 29 (2000), 415-430. doi: 10.1080/00411450008205882.

[28]

L. Pareschi and G. Russo, Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput., 23 (2001), 1253-1273. doi: 10.1137/S1064827500375916.

[29]

P.-A. Raviart, "An Analysis of Particle Methods in Numerical Methods in Fluid Dynamics," Lecture Notes in Math. 1127, F. Brezzi ed., Springer-Verlag, Berlin, 1985.

[30]

E. Sonnendrücker, Mathematical models for fusion,, Lecture Notes, (). 

[31]

B. Yan and S. Jin, A successive penalty-based asymptotic-preserving scheme for kinetic equations,, submitted., (). 

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