# American Institute of Mathematical Sciences

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 and 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), 121-153. doi: 10.1081/TT-100105365. [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), 341-365. doi: 10.1007/s10915-009-9302-4. [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), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. [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}, (). [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), 221-251. [6] J. F. Bourgat, P. LeTallec, B. Perthame and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, Domain decomposition and Engineering, Contemporary Mathematics, AMS, 157 (1992), 377-398. [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), 953-973. doi: 10.1137/S0036142997322102. [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), 113-149. doi: 10.1007/s10915-007-9181-5. [9] S. Chandrasekhar, "Radiative Transfer," Dover Publications, New-York, 1960. [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] N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equations, J. Comput. Phys., 203 (2005), 572-601. doi: 10.1016/j.jcp.2004.09.006. [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), 5630-5652. doi: 10.1016/j.jcp.2010.04.001. [13] P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comput. Phys., 229 (2010), 4907-4933. doi: 10.1016/j.jcp.2010.03.009. [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), 940-979. doi: 10.1137/060651574. [15] P. Degond and B. Lucquin-Desreux, An entropy scheme for the Fokker-Planck collision operator in the Coulomb case, Numer. Math., 68 (1994), 239-262. doi: 10.1007/s002110050059. [16] R. Duclous, B. Dubroca and F. Filbet, Analysis of a high order finite volume scheme for the Vlasov-Poisson system,, preprint.\vspace*{2pt}, (). [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. 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), 1033-1061. doi: 10.1007/BF02179863. [19] L. Gosse and G. Toscani, Asymptotic-preserving and well-balanced schemes for radiative transfer and the Rosseland approximation, Numer. Math., 98 (2004), 223-250. doi: 10.1007/s00211-004-0533-x. [20] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [21] S. Jin and D. Levermore, The discrete-ordinate method in diffusive regimes, Transport Theory Stat. Phys., 22 (1993), 739-791. doi: 10.1080/00411459308203842. [22] S. Jin and D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 126 (1996), 449-467. doi: 10.1006/jcph.1996.0149. [23] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num. Anal., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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), 4580-4606. doi: 10.1137/090756077. [25] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductors equations, SIAM J. Numer. Anal., 19 (1998), 2032-2050. [26] A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094. doi: 10.1137/S0036142996305558. [27] A. Klar, A numerical method for kinetic semiconductor equations in the drift diffusion limit, SIAM J. Sci. Comp., 20 (1999), 1696-1712. doi: 10.1137/S1064827597319258. [28] A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci., 11 (2001), 749-767. doi: 10.1142/S0218202501001082. [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), 891-913. doi: 10.1137/S0036142900375700. [30] M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. [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), 334-368. doi: 10.1137/07069479X. [32] T.-P. Liu and S.-H. Yu, Boltzmann equation: Micromacro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2. [33] G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes, Appl. Math. Lett., 11 (1998), 29-35. doi: 10.1016/S0893-9659(98)00006-8. [34] J. C. Mandal and S. M. Deshpande, Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23 (1994), 447-478. doi: 10.1016/0045-7930(94)90050-7. [35] J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [36] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation, J. Asympt. Anal., 4 (1991), 293-317.

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##### 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), 121-153. doi: 10.1081/TT-100105365. [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), 341-365. doi: 10.1007/s10915-009-9302-4. [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), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. [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}, (). [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), 221-251. [6] J. F. Bourgat, P. LeTallec, B. Perthame and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, Domain decomposition and Engineering, Contemporary Mathematics, AMS, 157 (1992), 377-398. [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), 953-973. doi: 10.1137/S0036142997322102. [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), 113-149. doi: 10.1007/s10915-007-9181-5. [9] S. Chandrasekhar, "Radiative Transfer," Dover Publications, New-York, 1960. [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] N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the Vlasov-BGK equations, J. Comput. Phys., 203 (2005), 572-601. doi: 10.1016/j.jcp.2004.09.006. [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), 5630-5652. doi: 10.1016/j.jcp.2010.04.001. [13] P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comput. Phys., 229 (2010), 4907-4933. doi: 10.1016/j.jcp.2010.03.009. [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), 940-979. doi: 10.1137/060651574. [15] P. Degond and B. Lucquin-Desreux, An entropy scheme for the Fokker-Planck collision operator in the Coulomb case, Numer. Math., 68 (1994), 239-262. doi: 10.1007/s002110050059. [16] R. Duclous, B. Dubroca and F. Filbet, Analysis of a high order finite volume scheme for the Vlasov-Poisson system,, preprint.\vspace*{2pt}, (). [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. 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), 1033-1061. doi: 10.1007/BF02179863. [19] L. Gosse and G. Toscani, Asymptotic-preserving and well-balanced schemes for radiative transfer and the Rosseland approximation, Numer. Math., 98 (2004), 223-250. doi: 10.1007/s00211-004-0533-x. [20] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [21] S. Jin and D. Levermore, The discrete-ordinate method in diffusive regimes, Transport Theory Stat. Phys., 22 (1993), 739-791. doi: 10.1080/00411459308203842. [22] S. Jin and D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 126 (1996), 449-467. doi: 10.1006/jcph.1996.0149. [23] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Num. Anal., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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), 4580-4606. doi: 10.1137/090756077. [25] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductors equations, SIAM J. Numer. Anal., 19 (1998), 2032-2050. [26] A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094. doi: 10.1137/S0036142996305558. [27] A. Klar, A numerical method for kinetic semiconductor equations in the drift diffusion limit, SIAM J. Sci. Comp., 20 (1999), 1696-1712. doi: 10.1137/S1064827597319258. [28] A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci., 11 (2001), 749-767. doi: 10.1142/S0218202501001082. [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), 891-913. doi: 10.1137/S0036142900375700. [30] M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. [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), 334-368. doi: 10.1137/07069479X. [32] T.-P. Liu and S.-H. Yu, Boltzmann equation: Micromacro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2. [33] G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes, Appl. Math. Lett., 11 (1998), 29-35. doi: 10.1016/S0893-9659(98)00006-8. [34] J. C. Mandal and S. M. Deshpande, Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23 (1994), 447-478. doi: 10.1016/0045-7930(94)90050-7. [35] J. Nieto, F. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158 (2001), 29-59. doi: 10.1007/s002050100139. [36] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation, J. Asympt. Anal., 4 (1991), 293-317.
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