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A Posteriori Stabilized Sixth-Order Finite Volume Scheme with Adaptive Stencil Construction: Basics for the 1D Steady-State Hyperbolic Equations

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Abstract

We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. 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. We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation. The stencil is shifted away from troubles (shocks, discontinuities, etc.) leading to less oscillating polynomial reconstructions. Experimented on linear, Bürgers’, and Euler equations, we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations. Moreover, we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.

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Acknowledgements

S. Clain and G.J. Machado acknowledge the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operational Fatores de Competitividade, and the National Funds through FCT – Fundação para a Ciência e a Tecnologia, project no. UID/FIS/04650/2019. S. Clain and G.J. Machado acknowledge the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operacional Fatores de Competitividade, and the National Funds through FCT – Fundação para a Ciência e a Tecnologia, project no. POCI-01-0145-FEDER-028118. The material of this research has been partly built and discussed during the SHARK workshops taking place in Póvoa de Varzim, Portugal, http://www.SHARK-FV.eu/.

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Authors and Affiliations

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

    Gaspar J. Machado & Stéphane Clain

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

    Raphaël Loubère

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

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Machado, G.J., Clain, S. & Loubère, R. A Posteriori Stabilized Sixth-Order Finite Volume Scheme with Adaptive Stencil Construction: Basics for the 1D Steady-State Hyperbolic Equations. Commun. Appl. Math. Comput. 5, 751–775 (2023). https://doi.org/10.1007/s42967-021-00140-7

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  • DOI: https://doi.org/10.1007/s42967-021-00140-7

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