Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus
<|reference_start|>Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus: We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincar\'e principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.<|reference_end|>
@article{carlier2024variational,
    title={Variational discretizations of ideal magnetohydrodynamics in smooth
  regime using finite element exterior calculus},
    author={Valentin Carlier and Martin Campos-Pinto},
    journal={arXiv preprint arXiv:2402.02905},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.02905},
    primaryClass={math.NA cs.NA}
}