A hierarchy of diffusion models for partially ionized plasmas

Choquet, Isabelle; Degond, Pierre; Lucquin-Desreux, Brigitte

doi:10.3934/dcdsb.2007.8.735

Cited by 14 publications

(8 citation statements)

References 19 publications

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“…We use an admissible change of variable which follows from the classical notion of entropy solutions in the context of first-order and second-order PDE. Entropy solutions have been found to play a fundamental role in the mathematical analysis of PDE [1,35], the analysis of complex physical systems [11,17], and the numerical approximation of PDE [27]. We also refer to the text-books by Dafermos [15] or Godlewski and Raviart [26].…”

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confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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To cite this version:Florent Renac, Claude Marmignon, Frédéric Coquel.Time implicit high-order discontinuous galerkin method with reduced evaluation cost.SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (1) Abstract. An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A Von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.Key words. discontinuous Galerkin method, nonlinear convection-diffusion equation, implicitexplicit time discretization, pseudo-time integration AMS subject classifications. 65N30, 65N121. Introduction. Discontinuous Galerkin (DG) methods are high-order finite element discretizations and were introduced in the early 1970s for the numerical simulation of the first-order hyperbolic neutron transport equation [34,40]. The method was later extended to nonlinear convection dominated flow problems with the use of a Runge-Kutta method for the time integration [12,14]. In recent years, there has been a strong interest for these techniques in the field of computational fluid dynamics which has led to the introduction of discretization schemes for parabolic and purely elliptic equations. See for example [4,5,8,9,13,19,20,24,28,37,43] and references cited therein. For more details, the reader is referred to the analysis of existing discretizations in an unified framework developed by Arnold et al. [3] and to the overview of recent progress in DG methods for compressible flows [32]. The success of these methods lies in their high-order of accuracy and flexibility thanks to their high degree of locality. The stencil of most DG methods is compact and independent of the space discretization order. This means that the evaluation of the residual in a discretization element involves only this element and its immediate neighbours. The order of the numerical scheme depends on the degree of the approximated piecewise polynomials which can be easily increased. These properties make the DG method well suited to algorithm parallelization, hp-refinement, hp-multigrid, unstructured meshes, the applicati...

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confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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“…The specific form depends on many variables, notably on the temperatures of each species. Their relative magnitudes for an atmospheric thermal arc discharge and glow discharge with or without interaction with an electron flow may be seen in Choquet et al (2007). The value for a totally ionized plasma for several kinetic distributions appears in Chen (1983) and Sturrock (1994).…”

Section: Introductionmentioning

confidence: 97%

Mechanisms of conversion of kinetic into magnetic energy in partially ionized plasmas

Núñez

2008

Geophysical & Astrophysical Fluid Dynamics

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The processes involved in the transformation of kinetic into magnetic energy in partially ionized compressible plasmas are not limited to the well-known mechanism of transport and diffusion of single-fluid magnetohydrodynamics. The induction equation, in this new case, may be written for any mean of the ion and electron velocities, and it possesses a natural scaling parameter equal to the ratio of the electron and ion masses. This produces a hierarchy of sources able to generate a magnetic field. The first of these sources corresponds to the classical stretching of magnetic field lines by the flow and ohmic diffusion, and a compressional effect due to the flow of electromagnetic energy into the regions of higher density. A potentially important effect is caused by dragging of electrons by neutral particles through collisions; this term could dissipate as well as enhance the magnetic field depending on the geometry of the neutral flow. Finally, there exists a term analogous to the stretching one, this time involving the current density and with the opposite effect. Its presence implies that not only the magnetic field but also the current density grows at most exponentially with a growth rate bounded by the maximum of the strain matrix. While the effect of these nonclassical sources, acting alone, is in all probability extremely small, they could be relevant in the creation of seed magnetic fields for subsequent dynamo action.

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“…This process gives rise to a significant impact on the electrons and ions dynamics which are not precisely described by the sole action of an external electric field [12][13][14] . Consequently, for some weakly ionized plasmas holding electric currents some models would lead to a non reliable description of the system if collisional effects are not properly considered, even in situations close to thermodynamic equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Three species one-dimensional kinetic model for weakly ionized plasmas

2016

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A three species one-dimensional kinetic model is presented for a spatially homogeneous weakly ionized plasma (WIP) subjected to the action of a time varying electric field. Planar geometry is assumed, which means that the plasma dynamics evolves in the privileged direction of the field. The energy transmitted to the charges is be channelized to the neutrals thanks to collisions and impacting the plasma dynamics.Charge-charge interactions have been designed as a one-dimensional collision term equivalent to the Landau operator used for fully ionized plasmas. Charge-neutral collisions are modelled by a conservative drift-diffusion operator in the Dougherty's form. The resulting set of coupled drift-diffusion equations is solved with the stable and robust Propagator Integral Method (PIM). This method feasibility accounts for non-linear effects without appealing to linearisation or simplifications, providing conservative physically meaningful solutions. It is found that charge-neutral collisions exert a significant effect since a quite different plasma dynamics arises if compared to the collisionless limit. In addition, substantial differences in the system evolution are found for constant and temperature dependent collision frequencies cases.

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A hierarchy of diffusion models for partially ionized plasmas

Abstract: International audienc

Cited by 14 publications

References 19 publications

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Mechanisms of conversion of kinetic into magnetic energy in partially ionized plasmas

Three species one-dimensional kinetic model for weakly ionized plasmas

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