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

Jordan Hoffart, Matthias Maier, John N. Shadid, Ignacio Tomas

TL;DR

This work develops a structure-preserving, fully discrete scheme for the electrostatic Euler-Poisson equations with a constant magnetic field, leveraging a Strang operator split between a hyperbolic Euler subsystem and a stiff, magnetically coupled source update. The source step is handled implicitly via a PDE Schur complement, reducing at each time step to a Poisson-like problem, which enables stable time stepping across the highly multiscale regime characteristic of magnetic-drift limits. The discretization uses finite element spaces with a lumped inner product to preserve positivity, entropy inequalities, and global energy balance, and includes restart strategies to enforce the discrete Gauss law. Numerical experiments on isentropic vortices and diocotron instabilities demonstrate accurate convergence and correct multiscale behavior, validating the method as a stepping stone toward full Euler-Maxwell solvers in electrostatic and magnetic-drift regimes.

Abstract

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 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 that differ by over 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's 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.

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

TL;DR

This work develops a structure-preserving, fully discrete scheme for the electrostatic Euler-Poisson equations with a constant magnetic field, leveraging a Strang operator split between a hyperbolic Euler subsystem and a stiff, magnetically coupled source update. The source step is handled implicitly via a PDE Schur complement, reducing at each time step to a Poisson-like problem, which enables stable time stepping across the highly multiscale regime characteristic of magnetic-drift limits. The discretization uses finite element spaces with a lumped inner product to preserve positivity, entropy inequalities, and global energy balance, and includes restart strategies to enforce the discrete Gauss law. Numerical experiments on isentropic vortices and diocotron instabilities demonstrate accurate convergence and correct multiscale behavior, validating the method as a stepping stone toward full Euler-Maxwell solvers in electrostatic and magnetic-drift regimes.

Abstract

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 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 that differ by over 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's 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.

Paper Structure

This paper contains 28 sections, 12 theorems, 83 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The compressible Euler equations with forces
  4. The Lorentz force
  5. Time scales of the magnetic Euler-Poisson equations
  6. Hyperbolic time scale $T_{\text{h}}$
  7. Plasma oscillation time scale $T_{\text{p}}=\omega_{\text{p}}^{-1}$
  8. Cyclotron and diocotron time scales $T_{\text{c}}=\omega_{\text{c}}^{-1}$, $T_{\text{d}}=\omega_{\text{d}}^{-1}$
  9. Operator splitting and PDE Schur complement
  10. Operator splitting
  11. A PDE Schur complement
  12. A fully discrete structure-preserving discretization
  13. Finite element spaces
  14. Assumptions on the hyperbolic solver
  15. Conservation
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $\Psi(\boldsymbol u):\mathbb{R}^{d+2} \rightarrow \mathbb{R}$ be a function of the state satisfying the functional dependence $\Psi(\boldsymbol u):= \psi(\rho, e(\boldsymbol u))$, where $e(\boldsymbol u) := \frac{\mathcal{E}}{\rho} - \frac{|\boldsymbol m|^2}{2 \rho^2}$ is the specific internal e where $\nabla_{\boldsymbol u}$ is the gradient with respect to the state. In other words, the time

Figures (4)

  • Figure 1: Temporal snapshots of a schlieren plot of the density profile of the third mode diocotron instability test case. Reference computation with no restart on refinement level $r=9$ amounting to 12,582,912 dG degrees of freedom per component. Here, $t_f = 10$.
  • Figure 2: Temporal snapshots of a schlieren plot of the density profile of the fourth mode diocotron instability test case. Reference computation with no restart on refinement level $r=9$ amounting to 12,582,912 dG degrees of freedom per component. Here, $t_f = 10$.
  • Figure 3: Temporal snapshots of a schlieren plot of the density profile of the fifth mode diocotron instability test case. Reference computation with no restart on refinement level $r=9$ amounting to 12,582,912 dG degrees of freedom per component. Here, $t_f = 10$.
  • Figure 4: Theoretical versus computed growth rates for modes (a) 3, (b) 4, and (c) 5. Log scale on the $y$ axis. Numerical growth rates are computed from the numerical amplitudes by fitting an exponential curve to the data between the square brackets in the plots from (a) $t = 0.4$ to $t = 0.7$, (b) $t = 0.6$ to $t = 0.75$, (c) $t = 1.15$ to $t = 1.35$. The numerical growth rates and their deviations from the theoretical growth rate (a) $\gamma_3 \approx 0.772$, (b) $\gamma_4 \approx 0.911$, (c) $\gamma_5 \approx 0.683$ are given in (d).

Theorems & Definitions (29)

  • Lemma 2.1
  • Proof 1
  • Corollary 2.2
  • Definition 2.3: Magnetic Euler-Poisson equations
  • Remark 2.4: Gauß law
  • Lemma 2.5: Energy balance
  • Proof 2
  • Definition 3.1: Strang splitting
  • Lemma 3.2: Semi-discrete parabolic energy balance
  • Proof 3
  • ...and 19 more