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Jahresbericht der Deutschen Mathematiker-Vereinigung

, Volume 120, Issue 3, pp 153–219 | Cite as

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

  • Dominik Derigs
  • Gregor J. Gassner
  • Stefanie Walch
  • Andrew R. Winters
Survey Article
  • 63 Downloads

Abstract

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas.

Keywords

Computational physics Entropy conservation Entropy stability Ideal MHD equations Finite volume methods 

Notes

Acknowledgements

Dominik Derigs and Stefanie Walch acknowledge the support of the Bonn-Cologne Graduate School for Physics and Astronomy (BCGS), which is funded through the Excellence Initiative.

Gregor Gassner has been supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487.

Stefanie Walch thanks the Deutsche Forschungsgemeinschaft (DFG) for funding through the SPP 1573 “The physics of the interstellar medium” and the European Research Council under the European Community’s Framework Programme FP8 via the ERC Starting Grant RADFEEDBACK (project number 679852).

This work has been partially performed using the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) HPC cluster at the Regionales Rechenzentrum Köln (RRZK), University of Cologne, Germany. Research in theoretical astrophysics is carried out within the Collaborative Research Centre 956, sub-project C5, funded by the Deutsche Forschungsgemeinschaft (DFG). The software used in this work was developed in part by the DOE NNSA ASC- and DOE Office of Science ASCR-supported FLASH Center for Computational Science at the University of Chicago.

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© Deutsche Mathematiker-Vereinigung and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  • Dominik Derigs
    • 1
  • Gregor J. Gassner
    • 2
  • Stefanie Walch
    • 1
  • Andrew R. Winters
    • 2
  1. 1.I. Physikalisches InstitutUniversität zu KölnKölnGermany
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany

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