2018/02/19 by Marvin Bohm, Andrew R. Winters, Bohm, Marvin +9 · 3 citations
Earth and Planetary Sciences · Engineering · #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1802.07341
openalex publication_date 2018/02/19 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
The first paper of this series presents a discretely entropy stable\ndiscontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD)\nequations on three-dimensional curvilinear unstructured hexahedral meshes.\nCompared to other fluid dynamics systems such as the shallow water equations or\nthe compressible Navier-Stokes equations, the resistive MHD equations need\nspecial considerations because of the divergence-free constraint on the\nmagnetic field. For instance, it is well known that for the symmetrization of\nthe ideal MHD system as well as the continuous entropy analysis a\nnon-conservative term proportional to the divergence of the magnetic field,\ntypically referred to as the Powell term, must be included. As a consequence,\nthe mimicry of the continuous entropy analysis in the discrete sense demands a\nsuitable DG approximation of the non-conservative terms in addition to the\nideal MHD terms.\n This paper focuses on the resistive MHD equations: Our first contribution is\na proof that the resistive terms are symmetric and positive-definite when\nformulated in entropy space as gradients of the entropy variables. This enables\nus to show that the entropy inequality holds for the resistive MHD equations.\nThis continuous analysis is the key for our DG discretization and guides the\npath for the construction of an approximation that discretely mimics the\nentropy inequality, typically termed entropy stability. Our second contribution\nis a detailed derivation and analysis of the discretization on\nthree-dimensional curvilinear meshes. The discrete analysis relies on the\nsummation-by-parts property, which is satisfied by the DG spectral element\nmethod (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. Although the\ndivergence-free constraint is included in the non-conservative terms, the\nresulting method has no particular treatment of the magnetic field divergence\nerrors...\n