An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field

TL;DR

This paper develops an Asymptotic-Preserving scheme for the Euler equations under strong magnetic fields, accurately capturing both the full model and drift limit without restrictive time-step constraints.

Contribution

It introduces a novel elliptic equation for parallel velocity enabling a scheme that is stable and consistent across different regimes of magnetic field strength.

Findings

The scheme accurately approximates the Euler-Lorentz system for finite epsilon.

It remains stable and consistent as epsilon approaches zero.

Numerical tests confirm the scheme's AP and stability properties.

Abstract

This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force. When the magnetic field is large, the so-called drift-fluid approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system. This scheme gives rise to both a consistent approximation of the Euler-Lorentz model when epsilon is finite and a consistent approximation of the drift limit when epsilon tends to 0. Above all, it does not require any constraint on the space and time steps related to the small value of epsilon. Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability.