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Bound-Preserving High Order Finite Volume Schemes for Conservation Laws and Convection-Diffusion Equations Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Stationarity and Vorticity Preservation for the Linearized Euler Equations in Multiple Spatial Dimensions

Author : Wasilij Barsukow

Published in: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects

Publisher: Springer International Publishing

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Abstract

Stationary solutions are a prominent subset of solutions to hyperbolic systems of PDEs. Failure of numerical methods to maintain stationarity is easily visible which makes these solutions an important class. Consider finite volume schemes solving multi-d linearized Euler equations on equidistant Cartesian grids. We formulate conditions for a scheme to have stationary states that are discretizations of all analytic stationary states. Such schemes are termed stationarity preserving. Stationarity preservation for the linearized Euler equations is shown to be equivalent to vorticity preservation.

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previous chapter A Nonlinear Discrete Duality Finite Volume Scheme for Convection-Diffusion Equations
next chapter Goal-Oriented Error Analysis of a DG Scheme for a Second Gradient Elastodynamics Model
Footnotes
1
p and \(\mathbf {v}\) will be called pressure and velocity, though, strictly speaking, when compared to (5)–(7) there is a factor of \(\bar{\rho }\), and thus a unit of mass missing.
 
2
Note that in this paper indices never denote derivatives.
 
3
All such statements are understood “up to machine error”.
 
Literature
1.
Amadori, D., Gosse, L.: Error Estimates for Well-Balanced Schemes on Simple Balance Laws: One-Dimensional Position-Dependent Models. Springer, Berlin (2015)
2.
Barsukow, W.: Stationarity preserving schemes for multi-dimensional linear systems. (2017, submitted)
3.
Barsukow, W., Klingenberg, C.: Exact solution and a truly multidimensional godunov scheme for the acoustic equations. Submitted (2016)
4.
Jeltsch, R., Torrilhon, M.: On curl-preserving finite volume discretizations for shallow water equations. BIT Numer. Math. 46(1), 35–53 (2006)MathSciNetCrossRefMATH
5.
Mishra, S., Tadmor, E.: ConstrainT Preserving Schemes Using Potential-based Fluxes ii. Genuinely Multi-dimensional Central Schemes for Systems of Conservation Laws. Preprint (2009)
6.
Morton, K.W., Roe, P.L.: Vorticity-preserving lax-wendroff-type schemes for the system wave equation. SIAM J. Sci. Comput. 23(1), 170–192 (2001)
7.
Torrilhon, M., Fey, M.: Constraint-preserving upwind methods for multidimensional advection equations. SIAM J. Numer. Anal. 42(4), 1694–1728 (2004)MathSciNetCrossRefMATH
8.
Ulrik, S.: Fjordholm and Siddhartha Mishra. Vorticity preserving finite volume schemes for the shallow water equations. SIAM J. Sci. Comput. 33(2), 588–611 (2011)MathSciNetCrossRefMATH
Metadata
Title
Stationarity and Vorticity Preservation for the Linearized Euler Equations in Multiple Spatial Dimensions
Author
Wasilij Barsukow
Copyright Year
2017
Book
Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects

Print ISBN: 978-3-319-57396-0

Electronic ISBN: 978-3-319-57397-7

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-57397-7_38

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