# American Institute of Mathematical Sciences

December  2011, 4(4): 991-1023. doi: 10.3934/krm.2011.4.991

## Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model

 1 Université de Bordeaux 1, 351, cours de la Libération, 33405 TALENCE Cedex, France 2 Institut de Mathématiques de Toulouse, 118, route de Narbonne, 31062 TOULOUSE Cedex, France, France 3 Laboratoire Paul Painlevé, UFR de Mathématiques, Cité Scientiﬁque, 59655 VILLENEUVE D’ASCQ Cedex, France

Received  April 2011 Revised  July 2011 Published  November 2011

The present work is devoted to the simulation of a strongly magnetized plasma considered as a mixture of an ion fluid and an electron fluid. For the sake of simplicity, we assume that the model is isothermal and described by Euler equations coupled with a term representing the Lorentz force. Moreover we assume that both Euler systems are coupled through a quasi-neutrality constraint of the form $n_{i}=n_{e}$. The numerical method which is described in the present document is based on an Asymptotic-Preserving semi-discretization in time of a variant of this two-fluid Euler-Lorentz model with a small perturbation of the quasi-neutrality constraint. Firstly, we present the two-fluid model and the motivations for introducing a small perturbation into the quasi-neutrality equation, then we describe the time semi-discretization of the perturbed model and a fully-discrete finite volume scheme based on it. Finally, we present some numerical results which have been obtained with this method.
Citation: Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic and Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991
##### References:
 [1] M. Beer and G. Hammett, Toroidal gyrofluid equations for simulations of tokamak turbulence, Phys. Plasmas, 3 (1996), 4046-4064. doi: 10.1063/1.871538. [2] R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendrücker, An asymptotically stable semi-lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., 41 (2009), 341-365. doi: 10.1007/s10915-009-9302-4. [3] A. Brizard, "Nonlinear Gyrokinetic Tokamak Physics," Ph.D thesis, Princeton University, 1990. [4] A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. [5] S. Brull, P. Degond and F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasmas simulations,, to appear in Commun. Comput. Phys., (). [6] C. Buet, S. Cordier, B. Lucquin-Desreux and S. Mancini, Diffusion limit of the Lorentz model: Asymptotic-preserving schemes, M2AN Math. Model. Numer. Anal. 36 (2002), 631-655. doi: 10.1051/m2an:2002028. [7] C. Buet and B. Després, Asymptotic-Preserving and positive schemes for radiation hydrodynamics, J. Comput. Phys., 215 (2006), 717-740. doi: 10.1016/j.jcp.2005.11.011. [8] J.-A. Carrillo, T. Goudon and P. Lafitte, Simulation of fluid and particles flows: Asymptotic-preserving schemes for bubbling and flowing regimes, J. Comput. Phys., 227 (2008), 7929-7951. doi: 10.1016/j.jcp.2008.05.002. [9] P. Crispel, P. Degond and M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, J. Comput. Phys., 205 (2005), 408-438. doi: 10.1016/j.jcp.2004.11.011. [10] P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208-234. doi: 10.1016/j.jcp.2006.09.004. [11] P. Crispel, P. Degond and M.-H. Vignal, A plasma expansion model based on the full Euler-Poisson system, Math. Models Methods Appl. Sci., 17 (2007), 1129-1158. doi: 10.1142/S0218202507002224. [12] 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, Kinet. Relat. Models, 4 (2011), 441-447. doi: 10.3934/krm.2011.4.441. [13] P. Degond, F. Deluzet, A. Lozinski, J. Narski and C. Negulescu, Duality-based asymptotic-preserving method for highly anisotropic diffusion equations,, to appear in Comm. Math. Sci., (). [14] 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. [15] P. Degond, F. Deluzet and C. Negulescu, An Asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul. 8 (2009/10), 645-666. doi: 10.1137/090754200. [16] P. Degond, F. Deluzet, A. Sangam and M.-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228 (2009), 3540-3558. doi: 10.1016/j.jcp.2008.12.040. [17] P. Degond, H. Liu, D. Savelief and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasi-neutral limit,, to appear in J. Sci. Comput., (). [18] P. Degond, J.-G. Liu and M.-H. Vignal, Analysis of an asymptotic-preserving scheme for the Euler-Poisson system in the quasi-neutral limit, J. Numer. Anal., 46 (2008), 1298-1322. doi: 10.1137/070690584. [19] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equation, Commun. Comput. Phys., 10 (2011), 1-31. [20] W. D. D'haeseleer, W. N. G. Hitchon, J. D. Callen and J. L. Shohet, "Flux Coordinates and Magnetic Field Structure. A Guide to a Fundamental Tool of Plasma Theory," Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. [21] W. Dorland and G. Hammett, Gyrofluid turbulence models with kinetic effects, Phys. Fluids B, 5 (1993), 812-835. doi: 10.1063/1.860934. [22] D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [23] G.-L. Falchetto and M. Ottaviani, Effect of collisional zonal-flow damping on flux-driven turbulent transport, Phys. Rev. Lett., 92 (2004), 025002. doi: 10.1103/PhysRevLett.92.025002. [24] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [25] F. Filbet and S. Jin, An asymptotic-preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x. [26] E. Frénod and A. Mouton, Two-dimensional finite larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math.: Advances Appl., 4 (2010), 135-166. [27] X. Garbet, C. Bourdelle, G.-T. Hoang, P. Maget, S. Benkadda, P. Beyer, C. Figarella, I. Voitsekovitch, O. Agullo and N. Bian, Global simulations of ion turbulence with magnetic shear reversal, Phys. Plasmas, 8 (2001), 2793-2803. doi: 10.1063/1.1367320. [28] V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Gendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclacik and L. Villard, A drift-kinetic semi-lagrangian 4D code for ion turbulence simulation, J. Comput. Phys., 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023. [29] V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-Pradalier, P. Gendrih, N. Crouseilles, G. Latu, E. Sonnendrücker, N. Besse and P. Bertrand, Computing ITG turbulence with a full-f semi-lagrangian code, Comm. Nonlinear Sci. Numer. Simul., 13 (2008), 81-87. doi: 10.1016/j.cnsns.2007.05.016. [30] G.-W. Hammett, M.-A. Beer, W. Dorland, S.-C. Cowley and S.-A. Smith, Developments in the gyrofluid approach to tokamak turbulence simulations, Plasmas Phys. Control. Fusion, 35 (1993), 973-985. doi: 10.1088/0741-3335/35/8/006. [31] A. Hasegawa and K. Mima, Stationary spectrum of strong turbulence in magnetized nonuniform plasma, Phys. Rev. Lett., 39 (1977), 205-208. doi: 10.1103/PhysRevLett.39.205. [32] A. Hasegawa and M. Wakatani, Plasma edge turbulence, Phys. Rev. Lett., 50 (1983), 682-686. doi: 10.1103/PhysRevLett.50.682. [33] R.-D. Hazeltine and J.-D. Meiss, "Plasma Confinement," Dover Publications, 2003. [34] J. A. Heikkinen, S. J. Janhunen, T. P. Kiviniemi and F. Ogando, Full-f gyrokinetic method for particle simulation of tokamak transport, J. Comput. Phys., 227 (2008), 5582-5609. doi: 10.1016/j.jcp.2008.02.013. [35] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [36] A. Klar, An asymptotic-preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., 36 (2009), 1507-1527. doi: 10.1137/S0036142997321765. [37] W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. doi: 10.1063/1.864140. [38] W.-W. Lee, Gyrokinetic particle simulation model, J. Comput. Phys., 72 (1987), 243-269. doi: 10.1016/0021-9991(87)90080-5. [39] 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. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X. [40] R. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [41] R. G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053. [42] R. G. McLarren and R. B. Lowrie, The effects of slope limiting on asymptotic-preserving numerical methods for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 9711-9726. doi: 10.1016/j.jcp.2008.07.012. [43] K. Miyamoto, "Controlled Fusion and Plasma Physics," Chapman & Hall, 2007. [44] M. Ottaviani, An alternative approach to field-aligned coordinates for plasma turbulence simulations,, preprint, (). [45] M. Ottaviani and G. Manfredi, The gyro-radius scaling of ion thermal transport from global numerical simulations of ion temperature gradient driven turbulence, Phys. Plasmas, 6 (1999), 3267-3275. doi: 10.1063/1.873567. [46] V.-V. Rusanov, The calculation of the interaction of non-stationary shock waves and obstacles, J. Comp. Math. Phys., 1 (1961), 267-279.

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##### References:
 [1] M. Beer and G. Hammett, Toroidal gyrofluid equations for simulations of tokamak turbulence, Phys. Plasmas, 3 (1996), 4046-4064. doi: 10.1063/1.871538. [2] R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendrücker, An asymptotically stable semi-lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., 41 (2009), 341-365. doi: 10.1007/s10915-009-9302-4. [3] A. Brizard, "Nonlinear Gyrokinetic Tokamak Physics," Ph.D thesis, Princeton University, 1990. [4] A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. [5] S. Brull, P. Degond and F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasmas simulations,, to appear in Commun. Comput. Phys., (). [6] C. Buet, S. Cordier, B. Lucquin-Desreux and S. Mancini, Diffusion limit of the Lorentz model: Asymptotic-preserving schemes, M2AN Math. Model. Numer. Anal. 36 (2002), 631-655. doi: 10.1051/m2an:2002028. [7] C. Buet and B. Després, Asymptotic-Preserving and positive schemes for radiation hydrodynamics, J. Comput. Phys., 215 (2006), 717-740. doi: 10.1016/j.jcp.2005.11.011. [8] J.-A. Carrillo, T. Goudon and P. Lafitte, Simulation of fluid and particles flows: Asymptotic-preserving schemes for bubbling and flowing regimes, J. Comput. Phys., 227 (2008), 7929-7951. doi: 10.1016/j.jcp.2008.05.002. [9] P. Crispel, P. Degond and M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, J. Comput. Phys., 205 (2005), 408-438. doi: 10.1016/j.jcp.2004.11.011. [10] P. Crispel, P. Degond and M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys., 223 (2007), 208-234. doi: 10.1016/j.jcp.2006.09.004. [11] P. Crispel, P. Degond and M.-H. Vignal, A plasma expansion model based on the full Euler-Poisson system, Math. Models Methods Appl. Sci., 17 (2007), 1129-1158. doi: 10.1142/S0218202507002224. [12] 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, Kinet. Relat. Models, 4 (2011), 441-447. doi: 10.3934/krm.2011.4.441. [13] P. Degond, F. Deluzet, A. Lozinski, J. Narski and C. Negulescu, Duality-based asymptotic-preserving method for highly anisotropic diffusion equations,, to appear in Comm. Math. Sci., (). [14] 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. [15] P. Degond, F. Deluzet and C. Negulescu, An Asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul. 8 (2009/10), 645-666. doi: 10.1137/090754200. [16] P. Degond, F. Deluzet, A. Sangam and M.-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228 (2009), 3540-3558. doi: 10.1016/j.jcp.2008.12.040. [17] P. Degond, H. Liu, D. Savelief and M.-H. Vignal, Numerical approximation of the Euler-Poisson-Boltzmann model in the quasi-neutral limit,, to appear in J. Sci. Comput., (). [18] P. Degond, J.-G. Liu and M.-H. Vignal, Analysis of an asymptotic-preserving scheme for the Euler-Poisson system in the quasi-neutral limit, J. Numer. Anal., 46 (2008), 1298-1322. doi: 10.1137/070690584. [19] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equation, Commun. Comput. Phys., 10 (2011), 1-31. [20] W. D. D'haeseleer, W. N. G. Hitchon, J. D. Callen and J. L. Shohet, "Flux Coordinates and Magnetic Field Structure. A Guide to a Fundamental Tool of Plasma Theory," Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. [21] W. Dorland and G. Hammett, Gyrofluid turbulence models with kinetic effects, Phys. Fluids B, 5 (1993), 812-835. doi: 10.1063/1.860934. [22] D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [23] G.-L. Falchetto and M. Ottaviani, Effect of collisional zonal-flow damping on flux-driven turbulent transport, Phys. Rev. Lett., 92 (2004), 025002. doi: 10.1103/PhysRevLett.92.025002. [24] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [25] F. Filbet and S. Jin, An asymptotic-preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x. [26] E. Frénod and A. Mouton, Two-dimensional finite larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math.: Advances Appl., 4 (2010), 135-166. [27] X. Garbet, C. Bourdelle, G.-T. Hoang, P. Maget, S. Benkadda, P. Beyer, C. Figarella, I. Voitsekovitch, O. Agullo and N. Bian, Global simulations of ion turbulence with magnetic shear reversal, Phys. Plasmas, 8 (2001), 2793-2803. doi: 10.1063/1.1367320. [28] V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Gendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclacik and L. Villard, A drift-kinetic semi-lagrangian 4D code for ion turbulence simulation, J. Comput. Phys., 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023. [29] V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-Pradalier, P. Gendrih, N. Crouseilles, G. Latu, E. Sonnendrücker, N. Besse and P. Bertrand, Computing ITG turbulence with a full-f semi-lagrangian code, Comm. Nonlinear Sci. Numer. Simul., 13 (2008), 81-87. doi: 10.1016/j.cnsns.2007.05.016. [30] G.-W. Hammett, M.-A. Beer, W. Dorland, S.-C. Cowley and S.-A. Smith, Developments in the gyrofluid approach to tokamak turbulence simulations, Plasmas Phys. Control. Fusion, 35 (1993), 973-985. doi: 10.1088/0741-3335/35/8/006. [31] A. Hasegawa and K. Mima, Stationary spectrum of strong turbulence in magnetized nonuniform plasma, Phys. Rev. Lett., 39 (1977), 205-208. doi: 10.1103/PhysRevLett.39.205. [32] A. Hasegawa and M. Wakatani, Plasma edge turbulence, Phys. Rev. Lett., 50 (1983), 682-686. doi: 10.1103/PhysRevLett.50.682. [33] R.-D. Hazeltine and J.-D. Meiss, "Plasma Confinement," Dover Publications, 2003. [34] J. A. Heikkinen, S. J. Janhunen, T. P. Kiviniemi and F. Ogando, Full-f gyrokinetic method for particle simulation of tokamak transport, J. Comput. Phys., 227 (2008), 5582-5609. doi: 10.1016/j.jcp.2008.02.013. [35] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [36] A. Klar, An asymptotic-preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., 36 (2009), 1507-1527. doi: 10.1137/S0036142997321765. [37] W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. doi: 10.1063/1.864140. [38] W.-W. Lee, Gyrokinetic particle simulation model, J. Comput. Phys., 72 (1987), 243-269. doi: 10.1016/0021-9991(87)90080-5. [39] 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. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X. [40] R. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [41] R. G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053. [42] R. G. McLarren and R. B. Lowrie, The effects of slope limiting on asymptotic-preserving numerical methods for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), 9711-9726. doi: 10.1016/j.jcp.2008.07.012. [43] K. Miyamoto, "Controlled Fusion and Plasma Physics," Chapman & Hall, 2007. [44] M. Ottaviani, An alternative approach to field-aligned coordinates for plasma turbulence simulations,, preprint, (). [45] M. Ottaviani and G. Manfredi, The gyro-radius scaling of ion thermal transport from global numerical simulations of ion temperature gradient driven turbulence, Phys. Plasmas, 6 (1999), 3267-3275. doi: 10.1063/1.873567. [46] V.-V. Rusanov, The calculation of the interaction of non-stationary shock waves and obstacles, J. Comp. Math. Phys., 1 (1961), 267-279.
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