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 and a minimum principle of the specific entropy, as well as global properties, such as 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 solution of the source system. Therefore, it reduces to a single non-symmetric Poisson-like problem that is solvedfor 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 characterizedby the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales with a difference in excess of $\mathcal{O}(10^{10})$, 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 existinganalytical 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.