Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

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 mot...

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Bibliographic Details
Published in:Jahresbericht der Deutschen Mathematiker-Vereinigung 2018, Vol.120 (3), p.153-219
Main Authors: Derigs, Dominik, Gassner, Gregor J., Walch, Stefanie, Winters, Andrew R.
Format: Article
Language:English
Subjects:
Approximation
Computational fluid dynamics
Computational physics
Derivation
Entropy
Entropy conservation
Entropy stability
Finite volume methods
Fluid flow
Ideal MHD equa- tions
Magnetic fluids
Magnetic properties
Magnetohydrodynamics
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Numerical methods
Partial differential equations
Physical properties
Plasmas
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Summary: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.
ISSN:0012-0456
1869-7135
1869-7135
DOI:10.1365/s13291-018-0178-9