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Interplay Between Diffusion Anisotropy and Mesh Skewness in Hybrid High-Order Schemes Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Well-Balanced Discretisation for the Compressible Stokes Problem by Gradient-Robustness

Authors : Alexander Linke, Christian Merdon

Published in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Publisher: Springer International Publishing

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Abstract

Based on the novel concept of gradient-robustness a well-balanced and provably convergent scheme for the compressible Stokes equations is discussed. Gradient-robustness means that arbitrary gradient fields in the momentum balance are correctly balanced by the discrete pressure gradient if there is enough mass in the system to compensate the force. For low Mach numbers the scheme degenerates to a recent inf-sup stable and pressure-robust discretisation for the incompressible Stokes equations. Numerical examples illustrate the properties for nearly-hydrostatic low Mach number flows also for nonlinear equations of state.

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previous chapter On the Significance of Pressure-Robustness for the Space Discretization of Incompressible High Reynolds Number Flows
next chapter A Second Order Consistent MAC Scheme for the Shallow Water Equations on Non Uniform Grids
Literature
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Akbas, M., Gallouet, T., Gassmann, A., Linke, A., Merdon, C.: A gradient-robust well-balanced scheme for the compressible isothermal stokes problem (2019)
2.
Eymard, R., Gallouët, T., Herbin, R., Latché, J.C.: A convergent finite element-finite volume scheme for the compressible Stokes problem II: The isentropic case. Math. Comp. 79(270), 649–675 (2010)MathSciNetCrossRef
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Feireisl, E.: Mathematical models of incompressible fluids as singular limits of complete fluid systems. Milan J. Math. 78(2), 523–560 (2010)MathSciNetCrossRef
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Feireisl, E., Lukáčová-Medviďová, M., Nečasová, v., Novotný, A., She, B.: Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime. Multiscale Model. Simul. 16(1), 150–183 (2018). https://​doi.​org/​10.​1137/​16M1094233
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Gallouët, T., Herbin, R., Latché, J.C.: A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: the isothermal case. Math. Comp. 78(267), 1333–1352 (2009)
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Gallouët, T., Herbin, R., Latché, J.C., Maltese, D.: Convergence of the MAC scheme for the compressible stationary Navier–Stokes equations. Math. Comp. 87(311), 1127–1163 (2018)MathSciNetCrossRef
7.
John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)MathSciNetCrossRef
8.
Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg. 311, 304–326 (2016)MathSciNetCrossRef
Metadata
Title
Well-Balanced Discretisation for the Compressible Stokes Problem by Gradient-Robustness
Authors
Alexander Linke
Christian Merdon
Copyright Year
2020
Book
Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Print ISBN: 978-3-030-43650-6

Electronic ISBN: 978-3-030-43651-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-43651-3_8

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