Study of an asymptotic preserving scheme for the quasi neutral Euler–Boltzmann model in the drift regime

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.