Bohm, Marvin, Winters, Andrew R., Gassner, Gregor J., Derigs, Dominik ORCID: 0000-0002-9687-2035, Hindenlang, Florian and Saur, Joachim (2019). An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and Numerical Verification. Technical Report.

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Abstract

The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on three-dimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible Navier-Stokes equations, the resistive MHD equations need special considerations because of the divergence-free constraint on the magnetic field. For instance, it is well known that for the symmetrization of the ideal MHD system as well as the continuous entropy analysis a non-conservative term proportional to the divergence of the magnetic field, typically referred to as the Powell term, must be included. As a consequence, the mimicry of the continuous entropy analysis in the discrete sense demands a suitable DG approximation of the non-conservative terms in addition to the ideal MHD terms. This paper focuses on the resistive MHD equations: Our first contribution is a proof that the resistive terms are symmetric and positive-definite when formulated in entropy space as gradients of the entropy variables, which enables us to show that the entropy inequality holds for the resistive MHD equations. This continuous analysis is the key for our DG discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality, typically termed entropy stability. Our second contribution is a detailed derivation and analysis of the discretization on three-dimensional curvilinear meshes. The discrete analysis relies on the summation-by-parts property, which is satisfied by the DG spectral element method (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. Although the divergence- free constraint is included in the non-conservative terms, the resulting method has no particular treatment of the magnetic field divergence errors, which might pollute the solution quality. Our final contribution is the extension of the standard resistive MHD equations and our DG approximation with a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). As a conclusion to the first part of this series, we provide detailed numerical validations of our DGSEM method that underline our theoretical derivations. In addition, we show a numerical example where the entropy stable DGSEM demonstrates increased robustness compared to the standard DGSEM.

Item Type: Preprints, Working Papers or Reports (Technical Report)
Creators:
CreatorsEmailORCID
Bohm, MarvinUNSPECIFIEDUNSPECIFIED
Winters, Andrew R.UNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIED
Derigs, Dominikderigs@ph1.uni-koeln.deorcid.org/0000-0002-9687-2035
Hindenlang, FlorianUNSPECIFIEDUNSPECIFIED
Saur, JoachimUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-92767
Series Name at the University of Cologne: Technical report series. Center for Data and Simulation Science
Volume: 2019,5
Subjects: Data processing Computer science
Mathematics
Technology (Applied sciences)
Uncontrolled Keywords:
KeywordsLanguage
resistive magnetohydrodynamicsEnglish
entropy stabilityEnglish
discontinuous Galerkin spectral element methodEnglish
hyperbolic divergence cleaningEnglish
curvilinear hexahedral meshEnglish
summation-by-partsEnglish
Faculty: Central Institutions / Interdisciplinary Research Centers
Divisions: Central Institutions / Interdisciplinary Research Centers > Center for Data and Simulation Science
Language: English
Date: 25 January 2019
URI: http://kups.ub.uni-koeln.de/id/eprint/9276

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