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a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

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An Erratum to this article was published on 07 October 2017

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Abstract

We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

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Change history

  • 07 October 2017

    During typesetting, Figs. 8 and 21 got corrupted and the images shown in the online published version are not correct. The original publication was updated.

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Acknowledgments

The material of this research has been partly built during the SHARK workshops taking place in Ofir, Portugal, http://www.math.univ-toulouse.fr/SHARK-FV/. The authors would like to acknowledge the financial support of Campus France through the PHC Pessoa labeled 26922YH and entitled “Investigation on very high-order finite volume numerical schemes for fluid hydrodynamics simulation”. The research was also supported by the Research Center CMAT of the University of Minho, Portugal, with the Portuguese Funds from the ”Fundação para a Ciência e a Tecnologia”, through the Project PEstOE/MAT/UI0013/2014.

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Authors and Affiliations

  1. Centre of Mathematics, University of Minho, Campus of Azurém, 4800-058, Guimarães, Portugal

    Stéphane Clain & Gaspar J. Machado

  2. CNRS and Institut de Mathématiques de Bordeaux (IMB), Université de Bordeaux, Talence, France

    Raphaël Loubère

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  1. Stéphane Clain
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  2. Raphaël Loubère
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  3. Gaspar J. Machado
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Correspondence to Gaspar J. Machado.

Additional information

Communicated by: Helge Holden

The original version of this article was revised: During typesetting figures 8 and 21 got corrupted and showed erroneous graph lines. Both figures were updated.

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Clain, S., Loubère, R. & Machado, G.J. a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations. Adv Comput Math 44, 571–607 (2018). https://doi.org/10.1007/s10444-017-9556-6

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