Structure-preserving finite-element approximations of the magnetic Euler-Poisson equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908590985445376 |
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| author | Hoffart, Jordan Maier, Matthias Shadid, John N. Tomas, Ignacio |
| author_facet | Hoffart, Jordan Maier, Matthias Shadid, John N. Tomas, Ignacio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure-preserving finite-element approximations of the magnetic Euler-Poisson equations Hoffart, Jordan Maier, Matthias Shadid, John N. Tomas, Ignacio Numerical Analysis 65M22, 35L65, 35Q31 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. |
| title | Structure-preserving finite-element approximations of the magnetic Euler-Poisson equations |
| topic | Numerical Analysis 65M22, 35L65, 35Q31 |
| url | https://arxiv.org/abs/2510.11808 |