Numerical approximation of the Euler-Maxwell model in the quasineutral limit

We derive and analyze an Asymptotic-Preserving scheme for the Euler-Maxwell system in the quasi-neutral limit. We prove that the linear stability condition on the time-step is independent of the scaled Debye length @l when @l->0. Numerical validation performed on Riemann initial data and for a model plasma opening switch device show that the AP-scheme is convergent to the Euler-Maxwell solution when @Dx/@l->0 where @Dx is the spatial discretization. But, when @l/@Dx->0, the AP-scheme is consistent with the quasi-neutral Euler-Maxwell system. The scheme is also perfectly consistent with the Gauss equation. The possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage.

[1]  Jian-Guo Liu,et al.  Analysis of an Asymptotic Preserving Scheme for the Euler-Poisson System in the Quasineutral Limit , 2008, SIAM J. Numer. Anal..

[2]  Pierre Degond,et al.  Quasi-neutral fluid models for current-carrying plasmas , 2005 .

[3]  Shi Jin,et al.  A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources , 2009, J. Comput. Phys..

[4]  P. Colella,et al.  A Conservative Finite Difference Method for the Numerical Solution of Plasma Fluid Equations , 1999 .

[5]  Pierre Degond,et al.  An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit , 2007, J. Comput. Phys..

[6]  Saulo Alves de Araujo,et al.  A B , 1996 .

[7]  M. Slemrod.  Shadowing and the Plasma-Sheath Transition Layer , 2001, J. Nonlinear Sci..

[8]  C. W. Nielson,et al.  A multidimensional quasineutral plasma simulation model , 1978 .

[9]  R. J. Mason,et al.  Implicit moment PIC-hybrid simulation of collisional plasmas , 1983 .

[10]  Dennis W. Hewett,et al.  Extension of the streamlined Darwin model to quasineutral plasmas , 1994 .

[11]  Luc Mieussens,et al.  Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics , 2008, J. Comput. Phys..

[12]  J. N. Leboeuf,et al.  A fluid-ion and particle-electron model for low-frequency plasma instabilities , 1992 .

[13]  J. Denavit,et al.  Hybrid simulation of ion beams in background plasma , 1987 .

[14]  Valery Godyak,et al.  Solving the mathematical model of the electrode sheath in symmetrically driven RF discharges , 1994 .

[15]  Bruno Després,et al.  Asymptotic preserving and positive schemes for radiation hydrodynamics , 2006, J. Comput. Phys..

[16]  Ryan G. McClarren,et al.  The effects of slope limiting on asymptotic-preserving numerical methods for hyperbolic conservation laws , 2008, J. Comput. Phys..

[17]  Fabrice Deluzet,et al.  An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field , 2009, J. Comput. Phys..

[18]  Axel Klar.  An Asymptotic Preserving Numerical Scheme for Kinetic Equations in the Low Mach Number Limit , 1999 .

[19]  Pierre Degond,et al.  Polynomial upwind schemes for hyperbolic systems , 1999 .

[20]  Bruce I. Cohen,et al.  Implicit time integration for plasma simulation , 1982 .

[21]  N. N. Kuznetsov.  Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation , 1976 .

[22]  Marshall Slemrod,et al.  Quasi-Neutral Limit for Euler-Poisson System , 2001, J. Nonlinear Sci..

[23]  Stéphane Cordier,et al.  An asymptotic preserving scheme for hydrodynamics radiative transfer models , 2007, Numerische Mathematik.

[24]  R. J. Mason,et al.  An electromagnetic field algorithm for 2d implicit plasma simulation , 1987 .

[25]  W. Manheimer,et al.  Electrostatic Particle-in-Cell Simulation Technique for Quasineutral Plasma , 1997 .

[26]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[27]  Jeremiah Brackbill,et al.  An implicit method for electromagnetic plasma simulation in two dimensions , 1982 .

[28]  Fabrice Deluzet,et al.  An Asymptotic Preserving Scheme for Strongly Anisotropic Elliptic Problems , 2009, Multiscale Model. Simul..

[29]  Marie Hélène Vignal.  A Boundary Layer Problem for an Asymptotic Preserving Scheme in the Quasi-Neutral Limit for the Euler--Poisson System , 2010, SIAM J. Appl. Math..

[30]  Bruce I. Cohen,et al.  Direct implicit large time-step particle simulation of plasmas , 1983 .

[31]  N. A. Krall,et al.  Principles of Plasma Physics , 1973 .

[32]  D. Hoff,et al.  Error bounds for Glimm difference approximations for scalar conservation laws , 1985 .

[33]  Jeremiah Brackbill,et al.  An implicit moment electromagnetic plasma simulation in cylindrical coordinates , 1986 .

[34]  Hailiang Liu,et al.  Numerical Approximation of the Euler-Poisson-Boltzmann Model in the Quasineutral Limit , 2010, J. Sci. Comput..

[35]  Axel Klar,et al.  Asymptotic-preserving schemes for unsteady flow simulations , 2006 .

[36]  Pierre Degond,et al.  An Asymptotically Stable Semi-Lagrangian scheme in the Quasi-neutral Limit , 2009, J. Sci. Comput..

[37]  F. Deluzet.  Mathematical modeling of plasma opening switches , 2003 .

[38]  S. Mancini,et al.  DIFFUSION LIMIT OF THE LORENTZ MODEL: ASYMPTOTIC PRESERVING SCHEMES , 2002 .

[39]  Fabrice Deluzet,et al.  Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations , 2010, 1008.3405.

[40]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Multidimensional Systems , 2002 .

[41]  C. Munz,et al.  The approximation of two-fluid plasma flow with explicit upwind scemes , 1995 .

[42]  D. Hewett,et al.  Electromagnetic direct implicit plasma simulation , 1987 .

[43]  Céline Parzani,et al.  A one-dimensional model of plasma expansion , 2003 .

[44]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[45]  Uri Shumlak,et al.  Approximate Riemann solver for the two-fluid plasma model , 2003 .

[46]  B. Lucier.  Error Bounds for the Methods of Glimm, Godunov and LeVeque , 1985 .

[47]  J. R. Ockendon,et al.  Asymptotic matching of plasma and sheath in an active low pressure discharge , 1970, Journal of Plasma Physics.

[48]  Céline Parzani,et al.  Plasma Expansion in Vacuum: Modeling the Breakdown of Quasi Neutrality , 2003, Multiscale Model. Simul..

[49]  P. Degond,et al.  All speed scheme for the low Mach number limit of the isentropic Euler equations , 2009, 0908.1929.

[50]  Marshall Slemrod.  The Radio-Frequency Driven Plasma Sheath: Asymptotics and Analysis , 2003, SIAM J. Appl. Math..

[51]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[52]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[53]  Thierry Goudon,et al.  Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes , 2008, J. Comput. Phys..

[54]  Fabrice Deluzet,et al.  Asymptotic-Preserving Particle-In-Cell method for the Vlasov-Poisson system near quasineutrality , 2010, J. Comput. Phys..

[55]  P. Rambo.  Finite-grid instability in quasineutral hybrid simulations , 1995 .

[56]  Laurent Gosse,et al.  Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation , 2004, Numerische Mathematik.

[57]  Seung-Yeal Ha,et al.  Global Existence of Plasma Ion-Sheaths and Their Dynamics , 2003 .

[58]  Sylvie Fabre.  Stability analysis of the Euler-poisson equations , 1992 .