On convergence of numerical solutions for the compressible MHD system with exactly divergence-free magnetic field

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00562018" target="_blank" >RIV/67985840:_____/22:00562018 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1137/21M1431011" target="_blank" >https://doi.org/10.1137/21M1431011</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/21M1431011" target="_blank" >10.1137/21M1431011</a>

Result language
angličtina
Original language name
On convergence of numerical solutions for the compressible MHD system with exactly divergence-free magnetic field
Original language description
We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence-free property of magnetic field exactly. Our strategy utilizes the recent concepts of dissipative weak solutions and consistent approximations. First, we show the dissipative weak-strong uniqueness principle, meaning a dissipative weak solution coincides with a classical solution as long as they emanate from the same initial data. Next, we show the convergence of consistent approximation toward the dissipative weak solution and thus the classical solution. Upon interpreting the consistent approximation as the stability and consistency of suitable numerical solutions we have established a generalized Lax equivalence theory: convergence - stability and consistency. Further, to illustrate the application of this theory, we propose two mixed finite volume-finite element methods with exact divergence-free magnetic field. Finally, by showing that solutions of these two schemes are consistent approximations, we conclude their convergence toward the dissipative weak solution and the classical solution.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA21-02411S" target="_blank" >GA21-02411S: Solving ill posed problems in the dynamics of compressible fluids</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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