Abstract
We deal with the numerical approximation of a simplified quasi neutral plasma model in the drift regime. Specifically, we analyze a finite volume scheme for the quasi neutral Euler–Boltzmann equations. We prove the unconditional stability of the scheme and give some bounds on the numerical approximation that are uniform in the asymptotic parameter. The proof relies on the control of the positivity and the decay of a discrete energy. The severe non linearity of the scheme being the price to pay to get the unconditional stability, to solve it, we propose an iterative linear implicit scheme that reduces to an elliptic system. The elliptic system enjoys a maximum principle that enables to prove the conservation of the positivity under a CFL condition that does not involve the asymptotic parameter. The linear L2 stability analysis of the iterative scheme shows that it does not request the mesh size and time step to be smaller than the asymptotic parameter. Numerical illustrations are given to illustrate the stability and consistency of the scheme in the drift regime as well as its ability to compute correct shock speeds.
Highlights
This work is devoted to the construction of a numerical scheme for the simulation of the quasi-neutral Euler– Boltzmann system: such a model represents the evolution of an unmagnetized quasi-neutral plasma where electrons are adiabatic and their density obeys the Boltzmann relation [14, 24] and ions are submitted to the self-generated electrostatic field
When ε → 0, we reach an asymptotic regime which is called the drift regime: in this regime the pressure force is balanced by the electrostatic-field
We considered the numerical discretization of the quasi-neutral Euler–Boltzmann equations
Summary
This work is devoted to the construction of a numerical scheme for the simulation of the quasi-neutral Euler– Boltzmann system: such a model represents the evolution of an unmagnetized quasi-neutral plasma where electrons are adiabatic and their density obeys the Boltzmann relation [14, 24] and ions are submitted to the self-generated electrostatic field. Rigorous stability analysis and proof of convergence in the limit ε → 0 for complete physical models is often a difficult task In this respect, the present work is devoted to analyse some properties of a scheme for the quasi-neutral Euler–Boltzmann system. The constant parameters ε > 0 and γ > 1 denote respectively the thermal energy relatively to the ions kinetic energy and the adiabatic constant As it is, the system (2.1)–(2.5) is not a closed set of equations, we use the physical hypothesis of quasi-neutrality with adiabatic electrons [24]. The study of the convergence of the solutions to (Pε) toward the solutions of (P0) when ε → 0+ is a classical problem arising in fluid mechanics and more generally in the framework of the so called singular limits of hyperbolic systems [1, 19, 23]
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