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arXiv · 2402.02905

Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus

Valentin Carlier·Martin Campos-Pinto
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Abstract

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.

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Valentin Carlier, Martin Campos-Pinto. 2024-02-05. Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus. https://arxiv.org/abs/2402.02905

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