Viscous regularization of the MHD equations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913224743452672 |
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| author | Dao, Tuan Anh Lundgren, Lukas Nazarov, Murtazo |
| author_facet | Dao, Tuan Anh Lundgren, Lukas Nazarov, Murtazo |
| contents | Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_03929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Viscous regularization of the MHD equations Dao, Tuan Anh Lundgren, Lukas Nazarov, Murtazo Numerical Analysis 34A45 G.1.8 Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection. |
| title | Viscous regularization of the MHD equations |
| topic | Numerical Analysis 34A45 G.1.8 |
| url | https://arxiv.org/abs/2402.03929 |