An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit

This paper deals with the modeling of a plasma in the quasineutral limit using the two-fluid Euler-Poisson system. In this limit, explicit numerical schemes suffer from severe numerical constraints related to the small Debye length and large plasma frequency. Here, we propose an implicit scheme which reduces to a scheme for the quasineutral Euler model in the quasineutral limit. Such a property is referred to as ''asymptotic preservation''. One of the distinctive features of this scheme is that it has a comparable numerical cost to that of an explicit scheme: simply the Poisson equation is replaced by a different (but formally equivalent) elliptic problem. We present numerical simulations for two different one-dimensional test-cases. They confirm the expected stability of the scheme in the quasineutral limit. They also show that this scheme has some accuracy problems in the limit of small electron to ion mass ratio in reproducing the correct electron velocity. But this problem is already present in the results of the classical algorithm. Numerical simulations are also performed for a two-dimensional problem of a plasma expansion in vacuum between two electrodes.

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

[2]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[3]  F. Poupaud,et al.  Global Solutions to the Isothermal Euler-Poisson System with Arbitrarily Large Data , 1995 .

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

[5]  Giovanni Lapenta,et al.  Particle-in-cell simulation of glow discharges in complex geometries , 1995 .

[6]  P. Degond,et al.  ELECTRON TRANSPORT AND SECONDARY EMISSION IN A SURFACE OF SOLAR CELL , 2000 .

[7]  Céline Parzani,et al.  On Plasma Expansion in Vacuum , 2003 .

[8]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[9]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[10]  Mengu Cho,et al.  Dielectric charging processes and arcing rates of high voltage solararrays , 1991 .

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

[12]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

[13]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[14]  Robert J. Hoekstra,et al.  Two‐dimensional modeling of high plasma density inductively coupled sources for materials processing , 1994 .

[15]  M. Vignal,et al.  Un modèle d'expansion de plasma dans le vide , 2002 .

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

[17]  J. Benford,et al.  Dynamics of a virtual cathode oscillator driven by a pinched diode , 1986 .

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

[19]  J. Weiner,et al.  Fundamentals and applications , 2003 .

[20]  Thomas Westermann,et al.  Particle-in-cell simulations with moving boundaries—adaptive mesh generation , 1994 .

[21]  D. C. Barnes,et al.  Direct Implicit Plasma Simulation , 1985 .

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

[23]  Roberto Natalini,et al.  Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[24]  P. Colella,et al.  An Implicit-Explicit Eulerian Godunov Scheme for Compressible Flow , 1995 .

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

[26]  M. Gibbons,et al.  The Darwin Direct Implicit Particle-in-Cell (DADIPIC) Method for Simulation of Low Frequency Plasma Phenomena , 1995 .

[27]  S. S. Kim,et al.  A new unconditionally stable algorithm for steady-state fluid simulation of high density plasma discharge , 2001 .

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

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

[30]  E. Grenier,et al.  Quasineutral limit of an euler-poisson system arising from plasma physics , 2000 .

[31]  Stanley Osher,et al.  Level Set Methods , 2003 .

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

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

[34]  Robert J. Hoekstra,et al.  Two‐dimensional hybrid model of inductively coupled plasma sources for etching , 1993 .

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

[36]  Pierre Degond,et al.  Transport coefficients of plasmas and disparate mass binary gases , 1996 .

[37]  J. Humphries,et al.  Modeling ion extraction from a free-plasma surface with a flexible conformal mesh , 2005 .

[38]  D. Payan,et al.  Electrostatic behaviour of materials in a charging space environment , 2004, Proceedings of the 2004 IEEE International Conference on Solid Dielectrics, 2004. ICSD 2004..

[39]  R. J. Mason,et al.  Implicit moment particle simulation of plasmas , 1981 .

[40]  Pierre Degond,et al.  An asymptotically stable discretization for the Euler–Poisson system in the quasi-neutral limit , 2005 .

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

[42]  P. Degond,et al.  An asymptotic analysis of the one-dimensional Vlasov-Poisson system : the Child-Langmuir law , 1991 .

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

[44]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[45]  S. Cordier,et al.  Système Euler-Poisson non linéaire. Existence globale de solutions faibles entropiques , 1998 .

[46]  J. Brackbill,et al.  Multiple time scales , 1985 .

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

[48]  R. J. Mason,et al.  Hybrid and Collisional Implicit Plasma Simulation Models , 1985 .

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

[50]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[51]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

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

[53]  Matthew R. Gibbons,et al.  Characterization of the Darwin Direct Implicit Particle-in-Cell Method and Resulting Guidelines for Operation , 1997 .

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

[55]  Dana A. Knoll,et al.  An Improved Convection Scheme Applied to Recombining Divertor Plasma Flows , 1998 .

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

[57]  Pierre Degond,et al.  THE CHILD-LANGMUIR LAW IN THE KINETIC THEORY OF CHARGED-PARTICLES. PART 1, ELECTRON FLOWS IN VACUUM , 1996 .

[58]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

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

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

[61]  H. C. Yee,et al.  Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations. [in gas dynamics , 1983 .

[62]  Shu Wang,et al.  Quasineutral Limit of Euler–Poisson System with and without Viscosity , 2005 .

[63]  J. Denavit,et al.  Time-implicit fluid simulation of collisional plasmas , 1992 .

[64]  Homa Karimabadi,et al.  A new asynchronous methodology for modeling of physical systems: breaking the curse of courant condition , 2005 .

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

[66]  H. C. Yee,et al.  Explicit and Implicit Multidimensional Compact High-Resolution Shock-Capturing Methods , 1997 .

[67]  L YoungsD,et al.  Time-dependent multi-material flow with large fluid distortion. , 1982 .

[68]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[69]  P. M Lyster,et al.  Abstract of papers to appear in future issuesA fluid-ion and particle-electron model for low-frequency plasma instabilities , 1992 .

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

[71]  J. Denavit,et al.  Fluid and field algorithms for time-implicit plasma simulation , 1991 .

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

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

[74]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[75]  Scott E. Parker,et al.  Multi-scale particle-in-cell plasma simulation , 1991 .

[76]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

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

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

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

[80]  Pierre Crispel,et al.  Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul , 2004 .

[81]  R. Klein.  Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics , 1995 .

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