June  2011, 4(2): 441-477. doi: 10.3934/krm.2011.4.441

An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits

1. 

INRIA-Nancy Grand Est and IRMA, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France

2. 

CNRS and IRMAR, Université de Rennes 1, 263 Avenue du General Leclerc CS74205, 35042 Rennes cedex, France

Received  October 2010 Revised  February 2011 Published  April 2011

In this work, we extend the micro-macro decomposition based numerical schemes developed in [3] to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in [30], we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when $\varepsilon\rightarrow 0$, which makes it free from the usual diffusion constraint $\Delta t=O(\Delta x^2)$ in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.
Citation: Nicolas Crouseilles, Mohammed Lemou. An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits. Kinetic & Related Models, 2011, 4 (2) : 441-477. doi: 10.3934/krm.2011.4.441
References:
[1]

A. Arnold, J.-A. Carrillo, I. Gamba and C.-W. Shu, Low and high-field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck system,, Transport Theory Statist. Phys., 30 (2001), 121.  doi: 10.1081/TT-100105365.  Google Scholar

[2]

R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendr焎ker, An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit,, J. Sc. Comput., 41 (2009), 341.  doi: 10.1007/s10915-009-9302-4.  Google Scholar

[3]

M. Benoune, 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.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[4]

L. L. Bonilla and J. Soler, High-field limit of the Vlasov-Poisson-Fokker-Planck system for different perturbation methods,, \url{http://arxiv.org/abs/cond-mat/0007164}.\vspace*{2pt}, ().   Google Scholar

[5]

M. Bostan and T. Goudon, Electric high-electric field limit for the Vlasov-Maxwell-Fokker-Planck system,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 221.   Google Scholar

[6]

J. F. Bourgat, P. LeTallec, B. Perthame and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping,, Domain decomposition and Engineering, 157 (1992), 377.   Google Scholar

[7]

C. Buet and S. Cordier, Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation,, SIAM J. Numer. Anal., 36 (1998), 953.  doi: 10.1137/S0036142997322102.  Google Scholar

[8]

J. A. Carrillo, T. Goudon, P. Lafitte and F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations,, J. Sci. Comput., 36 (2008), 113.  doi: 10.1007/s10915-007-9181-5.  Google Scholar

[9]

S. Chandrasekhar, "Radiative Transfer,", Dover Publications, (1960).   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]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equations,, J. Comput. Phys., 203 (2005), 572.  doi: 10.1016/j.jcp.2004.09.006.  Google Scholar

[12]

P. Degond, F. Deluzet, L. Navoret, A-B. Sun and M-H.Vignal, Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality,, J. Comput. Phys., 229 (2010), 5630.  doi: 10.1016/j.jcp.2010.04.001.  Google Scholar

[13]

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

[14]

P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects,, SIAM J. Multiscale Modeling and Simulations, 5 (2006), 940.  doi: 10.1137/060651574.  Google Scholar

[15]

P. Degond and B. Lucquin-Desreux, An entropy scheme for the Fokker-Planck collision operator in the Coulomb case,, Numer. Math., 68 (1994), 239.  doi: 10.1007/s002110050059.  Google Scholar

[16]

R. Duclous, B. Dubroca and F. Filbet, Analysis of a high order finite volume scheme for the Vlasov-Poisson system,, preprint.\vspace*{2pt}, ().   Google Scholar

[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.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[18]

F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for a kinetic linear half space problem,, J. Stat. Phys., 80 (1995), 1033.  doi: 10.1007/BF02179863.  Google Scholar

[19]

L. Gosse and G. Toscani, Asymptotic-preserving and well-balanced schemes for radiative transfer and the Rosseland approximation,, Numer. Math., 98 (2004), 223.  doi: 10.1007/s00211-004-0533-x.  Google Scholar

[20]

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

[21]

S. Jin and D. Levermore, The discrete-ordinate method in diffusive regimes,, Transport Theory Stat. Phys., 22 (1993), 739.  doi: 10.1080/00411459308203842.  Google Scholar

[22]

S. Jin and D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms,, J. Comput. Phys., 126 (1996), 449.  doi: 10.1006/jcph.1996.0149.  Google Scholar

[23]

S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations,, SIAM J. Num. Anal., 38 (2000), 913.  doi: 10.1137/S0036142998347978.  Google Scholar

[24]

S. Jin and Y. Shi, A micro-macro decomposition based on asymptotic-preserving scheme for the multispecies Boltzmann equation,, SIAM J. Sci. Comp., 31 (2010), 4580.  doi: 10.1137/090756077.  Google Scholar

[25]

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductors equations,, SIAM J. Numer. Anal., 19 (1998), 2032.   Google Scholar

[26]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit,, SIAM J. Numer. Anal., 35 (1998), 1073.  doi: 10.1137/S0036142996305558.  Google Scholar

[27]

A. Klar, A numerical method for kinetic semiconductor equations in the drift diffusion limit,, SIAM J. Sci. Comp., 20 (1999), 1696.  doi: 10.1137/S1064827597319258.  Google Scholar

[28]

A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models,, Math. Models Methods Appl. Sci., 11 (2001), 749.  doi: 10.1142/S0218202501001082.  Google Scholar

[29]

A. Klar and A. Unterreiter, Uniform stability of a finite difference scheme for transport equations in diffusive regimes,, SIAM J. Numer. Anal., 40 (2001), 891.  doi: 10.1137/S0036142900375700.  Google Scholar

[30]

M. Lemou, Relaxed micro-macro schemes for kinetic equations,, Comptes Rendus Math\'ematique, 348 (2010), 455.  doi: 10.1016/j.crma.2010.02.017.  Google Scholar

[31]

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.  doi: 10.1137/07069479X.  Google Scholar

[32]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micromacro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[33]

G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes,, Appl. Math. Lett., 11 (1998), 29.  doi: 10.1016/S0893-9659(98)00006-8.  Google Scholar

[34]

J. C. Mandal and S. M. Deshpande, Kinetic flux vector splitting for Euler equations,, Comput. Fluids, 23 (1994), 447.  doi: 10.1016/0045-7930(94)90050-7.  Google Scholar

[35]

J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system,, Arch. Ration. Mech. Anal., 158 (2001), 29.  doi: 10.1007/s002050100139.  Google Scholar

[36]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation,, J. Asympt. Anal., 4 (1991), 293.   Google Scholar

show all references

References:
[1]

A. Arnold, J.-A. Carrillo, I. Gamba and C.-W. Shu, Low and high-field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck system,, Transport Theory Statist. Phys., 30 (2001), 121.  doi: 10.1081/TT-100105365.  Google Scholar

[2]

R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendr焎ker, An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit,, J. Sc. Comput., 41 (2009), 341.  doi: 10.1007/s10915-009-9302-4.  Google Scholar

[3]

M. Benoune, 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.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[4]

L. L. Bonilla and J. Soler, High-field limit of the Vlasov-Poisson-Fokker-Planck system for different perturbation methods,, \url{http://arxiv.org/abs/cond-mat/0007164}.\vspace*{2pt}, ().   Google Scholar

[5]

M. Bostan and T. Goudon, Electric high-electric field limit for the Vlasov-Maxwell-Fokker-Planck system,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 221.   Google Scholar

[6]

J. F. Bourgat, P. LeTallec, B. Perthame and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping,, Domain decomposition and Engineering, 157 (1992), 377.   Google Scholar

[7]

C. Buet and S. Cordier, Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation,, SIAM J. Numer. Anal., 36 (1998), 953.  doi: 10.1137/S0036142997322102.  Google Scholar

[8]

J. A. Carrillo, T. Goudon, P. Lafitte and F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations,, J. Sci. Comput., 36 (2008), 113.  doi: 10.1007/s10915-007-9181-5.  Google Scholar

[9]

S. Chandrasekhar, "Radiative Transfer,", Dover Publications, (1960).   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]

N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equations,, J. Comput. Phys., 203 (2005), 572.  doi: 10.1016/j.jcp.2004.09.006.  Google Scholar

[12]

P. Degond, F. Deluzet, L. Navoret, A-B. Sun and M-H.Vignal, Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality,, J. Comput. Phys., 229 (2010), 5630.  doi: 10.1016/j.jcp.2010.04.001.  Google Scholar

[13]

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

[14]

P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects,, SIAM J. Multiscale Modeling and Simulations, 5 (2006), 940.  doi: 10.1137/060651574.  Google Scholar

[15]

P. Degond and B. Lucquin-Desreux, An entropy scheme for the Fokker-Planck collision operator in the Coulomb case,, Numer. Math., 68 (1994), 239.  doi: 10.1007/s002110050059.  Google Scholar

[16]

R. Duclous, B. Dubroca and F. Filbet, Analysis of a high order finite volume scheme for the Vlasov-Poisson system,, preprint.\vspace*{2pt}, ().   Google Scholar

[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.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[18]

F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for a kinetic linear half space problem,, J. Stat. Phys., 80 (1995), 1033.  doi: 10.1007/BF02179863.  Google Scholar

[19]

L. Gosse and G. Toscani, Asymptotic-preserving and well-balanced schemes for radiative transfer and the Rosseland approximation,, Numer. Math., 98 (2004), 223.  doi: 10.1007/s00211-004-0533-x.  Google Scholar

[20]

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

[21]

S. Jin and D. Levermore, The discrete-ordinate method in diffusive regimes,, Transport Theory Stat. Phys., 22 (1993), 739.  doi: 10.1080/00411459308203842.  Google Scholar

[22]

S. Jin and D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms,, J. Comput. Phys., 126 (1996), 449.  doi: 10.1006/jcph.1996.0149.  Google Scholar

[23]

S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations,, SIAM J. Num. Anal., 38 (2000), 913.  doi: 10.1137/S0036142998347978.  Google Scholar

[24]

S. Jin and Y. Shi, A micro-macro decomposition based on asymptotic-preserving scheme for the multispecies Boltzmann equation,, SIAM J. Sci. Comp., 31 (2010), 4580.  doi: 10.1137/090756077.  Google Scholar

[25]

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductors equations,, SIAM J. Numer. Anal., 19 (1998), 2032.   Google Scholar

[26]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit,, SIAM J. Numer. Anal., 35 (1998), 1073.  doi: 10.1137/S0036142996305558.  Google Scholar

[27]

A. Klar, A numerical method for kinetic semiconductor equations in the drift diffusion limit,, SIAM J. Sci. Comp., 20 (1999), 1696.  doi: 10.1137/S1064827597319258.  Google Scholar

[28]

A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models,, Math. Models Methods Appl. Sci., 11 (2001), 749.  doi: 10.1142/S0218202501001082.  Google Scholar

[29]

A. Klar and A. Unterreiter, Uniform stability of a finite difference scheme for transport equations in diffusive regimes,, SIAM J. Numer. Anal., 40 (2001), 891.  doi: 10.1137/S0036142900375700.  Google Scholar

[30]

M. Lemou, Relaxed micro-macro schemes for kinetic equations,, Comptes Rendus Math\'ematique, 348 (2010), 455.  doi: 10.1016/j.crma.2010.02.017.  Google Scholar

[31]

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.  doi: 10.1137/07069479X.  Google Scholar

[32]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micromacro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[33]

G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes,, Appl. Math. Lett., 11 (1998), 29.  doi: 10.1016/S0893-9659(98)00006-8.  Google Scholar

[34]

J. C. Mandal and S. M. Deshpande, Kinetic flux vector splitting for Euler equations,, Comput. Fluids, 23 (1994), 447.  doi: 10.1016/0045-7930(94)90050-7.  Google Scholar

[35]

J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system,, Arch. Ration. Mech. Anal., 158 (2001), 29.  doi: 10.1007/s002050100139.  Google Scholar

[36]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation,, J. Asympt. Anal., 4 (1991), 293.   Google Scholar

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