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

We develop a structure-preserving numerical discretization for the electrostatic Euler-Poisson equations with a constant magnetic field. The scheme preserves positivity of the density, positivity of the internal energy, a minimum principle for the specific entropy, and a total energy balance. The scheme uses an operator splitting approach composed of two subsystems: the compressible Euler equations of gas dynamics, and a source system. The source system couples the electrostatic potential, momentum, and Lorentz force, thus incorporating electrostatic plasma and cyclotron motions. Because of the high-frequency phenomena it describes, the source system is discretized with an implicit time-stepping scheme. We use a PDE Schur complement approach for the numerical approximation of the source system, reducing to a single non-symmetric Poisson-like problem that is solved for each time step. Our focus with the present work is on the efficient solution of problems close to the magnetic-drift limit. Such asymptotic limit is characterized by the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales with a difference in excess of 10 orders of magnitude, making their numerical solution quite challenging. 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.