Structure-preserving finite-element approximations of the magnetic Euler-Poisson equations

Abstract

We develop a structure-preserving numerical discretization for the electrostatic Euler-Poisson equations with a given magnetic field. Our focus is on the efficient solution of problems close to the magnetic-drift limit. This regime is characterized by the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales with a difference in excess of 12 orders of magnitude. Our scheme oversteps these high-frequency oscillations with a time-step size restricted only by a hyperbolic CFL condition, while also preserving positivity of the density, positivity of the internal energy, a minimum principle for the specific entropy, and a total energy balance. The scheme uses a semi-implicit operator splitting approach composed of two subsystems: an explicit hyperbolic update with the compressible Euler equations of gas dynamics, and an implicit source update that couples the electrostatic potential, momentum, and Lorentz force. The Lorentz force is eliminated pointwise from the source system by a PDE Schur complement, so that only a single scalar, non-symmetric Poisson-like problem has to be solved per source update, which in turn enables the use of matrix-free linear algebra. Because of the efficiency of the matrix-free linear algebra, the implicit source solve costs about as much as a single explicit hyperbolic update, and accounts for less than a third of the total wall time. We illustrate the capability of the scheme by computing a diocotron instability and present growth rates that compare favorably with existing analytical results. The model, though a simplified version of the Euler-Maxwell system, represents a stepping stone toward electromagnetic solvers that are capable of working in the electrostatic and magnetic-drift limits as well as the hydrodynamic regime.