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Numerical methods for plasma physics in collisional regimes

  • G. Dimarco (a1), Q. Li (a2), L. Pareschi (a1) and B. Yan (a3)
Abstract

We consider the development of accurate and efficient numerical methods for the solution of the Vlasov–Landau equation describing a collisional plasma. The methods combine a Lagrangian approach for the Vlasov solver with a fast spectral method for the solution of the Landau operator. To this goal, new modified spectral methods for the Landau integral which are capable to capture correctly the Maxwellian steady state are introduced. A particular care is devoted to the construction of Implicit–Explicit and Exponential Runge–Kutta methods that permit to achieve high-order and efficient time integration of the collisional step. Several numerical tests are reported which show the high accuracy of the numerical schemes here presented.

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Email address for correspondence: giacomo.dimarco@unife.it
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Alexandre, R. and Villani, C. 2004 On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21 (1), 6195.
Arsenév, A. A. 1989 On a connection between the Boltzmann equation and the Landau-Fokker-Planck equations. Dokl. Akad. Nauk SSSR 305 (2), 322324.
Ascher, U. M., Ruuth, S. J. and Spiteri, R. J. 1997 Implicit-explicit runge-kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151167.
Ayuso, B., Carrillo, J. A. and Shu, C.-W. 2011 Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinet. Relat. Models 4 (4), 955989.
Bobylev, A. and Rjasanow, S. 1997 Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B Fluids 16 (2), 293306.
Bobylev, A. V. and Nanbu, K 2000 Theory of collision algorithms for gases and plasmas based on the boltzmann equation and the landau-fokker-planck equation. Phys. Rev. E 61, 45764586.
Bobylev, A. V., Potapenko, I. F. and Chuyanov, V. A. 1980 Completely conservative difference schemes for nonlinear kinetic equations of Landau (Fokker-Planck) type. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 26.
Bobylev, A. V. and Rjasanow, S. 1999 Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B Fluids 18 (5), 869887.
Boris, J. P. and Book, D. L. 1973 Flux-corrected transport I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11, 3869.
Buet, C. and Cordier, S. 1998 Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau equation. J. Comput. Phys. 145 (1), 228245.
Buet, C. and Cordier, S. 1999 Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation. SIAM J. Numer. Anal. 36 (3), 953973 (electronic).
Buet, C., Cordier, S., Degond, P. and Lemou, M. 1997 Fast algorithms for numerical, conservative, and entropy approximations of the Fokker-Planck-Landau equation. J. Comput. Phys. 133 (2), 310322.
Buet, C., Cordier, S. and Filbet, F. 1999 Comparison of numerical schemes for Fokker-Planck-Landau equation. In: CEMRACS 1999 (Orsay), Proc. ESAIM, Vol. 10, Soc. Math. Appl. Indust., Paris, pp. 161–181 (electronic).
Caflisch, R., Jin, S. and Russo, G. 1997 Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1), 246281.
Caflisch, R., Wang, C., Dimarco, G., Cohen, B. and Dimits, A. 2008 A hybrid method for accelerated simulation of Coulomb collisions in a plasma. Multiscale Model. Simul. 7 (2), 865887.
Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A. 1988 Spectral Methods in Fluid Dynamics, New York: Springer-Verlag.
Carrillo, J. A. and Vecil, F. 2007 Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 29, 11791206.
Cercignani, C. 1988 The Boltzmann Equation and its Applications (Applied Mathematical Sciences, 67). New York: Springer-Verlag.
Cercignani, C., Illner, R. and Pulvirenti, M. 1994 The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences, 106). New York: Springer-Verlag.
Cheng, C. Z. and Knorr, G. 1976 The integration of the Vlasov equation in configuration space. Comput. Phys. Commun. 22, 330335.
Cheng, Y., Gamba, I. M. and Proft, J. 2012 Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations. Math. Comput. 81 (277), 153190.
Crouseilles, N., Gutnic, M., Latu, G. and Sonnendrücker, E. 2008 Comparison of two Eulerian solvers for the four-dimensional Vlasov equation. I. Commun. Nonlinear Sci. Numer. Simul. 13 (1), 8893.
Crouseilles, N., Mehrenberger, M. and Sonnendrücker, E. 2010 Conservative semi-lagrangian schemes for Vlasov-type equations. J. Comput. Phys. 229, 19271953.
Degond, P. 2014 Asymptotic-preserving schemes for fluid models of plasmas. Panoramas et Syntheses.
Degond, P. and Lucquin-Desreux, B. 1992 The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Models Methods Appl. Sci. 2 (2), 167182.
Degond, P. and Lucquin-Desreux, B. 1994 An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory. Numer. Math. 68 (2), 239262.
Desvillettes, L. 1992 On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Theory Stat. Phys. 21 (3), 259276.
Dia, B. O. and Schatzman, M. 1996 Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées. RAIRO Modél. Math. Anal. Numér. 30 (3), 343383.
Dimarco, G., Caflisch, R. and Pareschi, L. 2010 Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas. Commun. Appl. Ind. Math. 1 (1), 7291.
Dimarco, G. and Pareschi, L. 2011 Exponential Runge-Kutta methods for stiff kinetic equations. SIAM J. Numer. Anal. 49 (5), 20572077.
Dimarco, G. and Pareschi, L. 2012 High order asymptotic-preserving schemes for the Boltzmann equation. C. R. Math. 350 (910), 481486.
Dimarco, G. and Pareschi, L. 2013 Asymptotic preserving implicit-Explicit Runge-Kutta methods for nonlinear kinetic equations. SIAM J.Numer. Anal. 51 (2), 10641087.
Dimarco, G. and Pareschi, L. 2014 Numerical methods for kinetic equations. Acta Numer. 23, 369520.
Filbet, F. and Jin, S. 2010 A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229 (20), 76257648.
Filbet, F. and Mouhot, C. 2011 Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Am. Math. Soc. 363 (4), 19471980.
Filbet, F., Mouhot, C. and Pareschi, L. 2006 Solving the Boltzmann equation in N log2 N . SIAM J. Sci. Comput. 28 (3), 10291053 (electronic).
Filbet, F. and Pareschi, L. 2002 A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case. J. Comput. Phys. 179 (1), 126.
Filbet, F. and Pareschi, L. 2003 Numerical solution of the Fokker-Planck-Landau equation by spectral methods. Commun. Math. Sci. 1 (1), 206207.
Filbet, F., Pareschi, L. and Rey, T. 2014 On steady state preserving spectral methods for homogeneous Boltzmann equations. C. R. Acad. Sci. (submitted).
Filbet, F., Sonnendrücker, E. and Bertrand, P. 2001 Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172 (1), 166187.
Gamba, I. M. and Haack, J. R. 2014 A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit. J. Comput. Phys. 270, 4057.
Gamba, I. M. and Tharkabhushanam, S. H. 2009 Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228 (6), 20122036.
Ghanshyam, N. and Tripathi, V. K. 1993 Self-focusing and filamentation of laser beams in collisional plasmas with finite thermal conduction. J. Plasma Phys. 49, 243253.
Guo, W. and Qiu, J.-M. 2013 Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation. J. Comput. Phys. 234, 108132.
Hairer, E., Lubich, C. and Wanner, G. 2010 Geometric Numerical Integration (Springer Series in Computational Mathematics, 31). Springer, Heidelberg, structure-preserving algorithms for ordinary differential equations, Reprint of the second (2006) edition.
Heath, R. E., Gamba, I. M., Morrison, P. J. and Michler, C. 2012 A discontinuous Galerkin method for the Vlasov-Poisson system. J. Comput. Phys. 231 (4), 11401174.
Hochbruck, M. and Ostermann, A. 2010 Exponential integrators. Acta Numer. 19, 209286.
Hu, J., Li, Q. and Pareschi, L 2014 Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy. J. Sci. Comput. (to appear).
Jin, S. 1995 Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122 (1), 5167.
Jin, S. 1999 Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441454.
Jin, S 2012 Asymptotic preserving (ap) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Mat. Univ. Parma 3 177216.
Jin, S. and Yan, B. 2011 A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation. J. Comput. Phys. 230 (17), 64206437.
Khabibrakhmanov, I. K. and Khazanov, G. V. 2000 The spectral collocation method for the kinetic equation with the nonlinear two-dimensional coulomb collisional operator. J. Comput. Phys. 161, 558575.
Landau, L. D. 1936 Die kinetische gleichung für den fall coulombscher vechselwirkung (the transport equation in the case of the coulomb interaction). Phys. Z. Sowjet. 154.
Lemou, M. 1998 Multipole expansions for the Fokker-Planck-Landau operator. Numer. Math. 78 (4), 597618.
Li, Q. and Pareschi, L. 2014 Exponential Runge-Kutta for the inhomogeneous Boltzmann equations with high order of accuracy. J. Comput. Phys. 259, 402420.
Li, Q., Pareschi, L. and Yan, B. 2014 Efficient time integration of the Fokker-Planck-Landau equation. preprint.
Li, Q. and Yang, X. 2014 Exponential Runge-Kutta methods for the multi-species Boltzmann equation. Commun. Comput. Phys. 15, 9961011.
Moler, C. and Loan, C. V. 1978 Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801836.
Mouhot, C. and Pareschi, L. 2004 Fast methods for the Boltzmann collision integral. C. R. Math. Acad. Sci. Paris 339 (1), 7176.
Mouhot, C. and Pareschi, L. 2006 Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (256), 18331852 (electronic).
Pareschi, L. and Perthame, B. 1996 A Fourier spectral method for homogeneous Boltzmann equations. In: Proc. 2nd Int. Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), Vol. 25, pp. 369–382.
Pareschi, L. and Russo, G. 1999 An introduction to Monte Carlo methods for the Boltzmann equation. In: CEMRACS 1999 (Orsay), Proc. ESAIM, Vol. 10, Soc. Math. Appl. Indust., Paris, pp. 3576.
Pareschi, L. and Russo, G. 2000a Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (4), 12171245.
Pareschi, L. and Russo, G. 2000b On the stability of spectral methods for the homogeneous Boltzmann equation. In: Proc. 5th Int. Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), Vol. 29, pp. 431–447.
Pareschi, L. and Russo, G. 2005 Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (1–2).
Pareschi, L. and Russo, G. 2011 Efficient asymptotic preserving deterministic methods for the boltzmann equation. In: Models and Computational Methods for Rarefied Flows, AVT-194 RTO AVT/VKI. Rhode St. Genese, Belgium.
Pareschi, L., Russo, G. and Toscani, G. 2000a Fast spectral methods for the Fokker-Planck-Landau collision operator. J. Comput. Phys. 165 (1), 216236.
Pareschi, L., Russo, G. and Toscani, G. 2000b Méthode spectrale rapide pour l'équation de Fokker-Planck-Landau. C. R. Acad. Sci. Paris Sér. I Math. 330 (6), 517522.
Pareschi, L., Toscani, G. and Villani, C. 2003 Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93 (3), 527548.
Pitale, L. A. 1978 Filamentation of a laser beam in a strongly ionized magnetoplasma. J. Plasma Phys. 19, 5561.
Qiu, J.-M. and Shu, C.-W. 2011 Positivity preserving semi-lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the VlasovPoisson system. J. Comput. Phys. 230, 83868409.
Rosenbluth, M. N., MacDonald, W. M. and Judd, D. L. 1957 Fokker-Planck equation for an inverse-square force. Phys. Rev. 107 (2), 16.
Sonnendrücker, E. 2013 Numerical methods for Vlasov equations. Tech. Rep.. MPI TU Munich, (http://www-m16.ma.tum.de/foswiki/pub/M16/Allgemeines/NumMethVlasov/Num-Meth-Vlasov-Notes.pdf).
Sonnendrücker, E., Roche, J., Bertrand, P. and Ghizzo, A. 1999 The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149 (2), 201220.
Strang, G. 1968 On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506517.
Sydora, R. D., Detering, F., Rozmus, W., Bychenkov, Y. Yu., Brantov, A. and Capjack, C. E. 2006 Collisional particle simulation of ion acoustic instability. J. Plasma Phys. 72, 12951298.
Valentini, F., Onofri, M. and Primavera, L. 2009 Mixed finite difference-spectral numerical approach for kinetic and fluid description of nonlinear phenomena in plasma physics. In: Numerical Simulation Research Progress, New York: Nova Sci. Publ. pp. 99139.
Valentini, F., Trávníček, P., Califano, F., Hellinger, P. and Mangeney, A. 2007 A hybrid-Vlasov model based on the current advance method for the simulation of collisionless magnetized plasma. J. Comput. Phys. 225 (1), 753770.
Villani, C. 2002 A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, Vol. I, Amsterdam: North-Holland, pp. 71305.
Wang, C., Lin, T., Caflisch, R., Cohen, B. I. and Dimits, A. M. 2008 Particle simulation of Coulomb collisions: comparing the methods of Takizuka & Abe and Nanbu. J. Comput. Phys. 227 (9), 43084329.
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