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High-Order Arbitrary-Lagrangian-Eulerian Schemes on Crazy Moving Voronoi Meshes

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Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems (YR 2021)

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Abstract

Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination of space-time-based high-order schemes, very accurate on smooth features even on coarse grids, with Lagrangian methods, which, by moving the mesh with the fluid flow, yield highly resolved and minimally dissipative results on both shocks and contacts. However, ensuring the high quality of moving meshes is a huge challenge that needs the development of innovative and unconventional techniques. The scheme proposed here falls into the family of Arbitrary-Lagrangian-Eulerian (ALE) methods, with the unique additional freedom of evolving the shape of the mesh elements through connectivity changes. We aim here at showing, by simple and very salient examples, the capabilities of high-order ALE schemes, and of our novel technique, based on the high-order space-time treatment of topology changes.

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Acknowledgements

E. Gaburro is member of the CARDAMOM team at the Inria center of the University of Bordeaux in France and S. Chiocchetti is member of the INdAM GNCS group in Italy. E. Gaburro gratefully acknowledges the support received from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Individual Fellowship SuPerMan, grant agreement No. 101025563. S. Chiocchetti acknowledges the support obtained by the Deutsche Forschungsgemeinschaft (DFG) via the project DROPIT, grant no. GRK 2160/2, and from the European Union’s Horizon Europe Research and Innovation Programme under the Marie Skłodowska-Curie Postdoctoral Fellowship MoMeNTUM, grant agreement No. 101109532.

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Authors and Affiliations

  1. Inria, University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 200 Avenue de la Vieille Tour, 33405, Talence cedex, France

    Elena Gaburro

  2. Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano, 77, 38123, Trento, Italy

    Simone Chiocchetti

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  1. Elena Gaburro
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Correspondence to Elena Gaburro .

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  1. Department of Informatics, University of Verona, Verona, Italy

    Giacomo Albi

  2. Department of Mathematics and Informatics, University of Ferrara, Ferrara, Italy

    Walter Boscheri

  3. Department of Mathematics, University of Pavia, Pavia, Italy

    Mattia Zanella

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Gaburro, E., Chiocchetti, S. (2023). High-Order Arbitrary-Lagrangian-Eulerian Schemes on Crazy Moving Voronoi Meshes. In: Albi, G., Boscheri, W., Zanella, M. (eds) Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems. YR 2021. SEMA SIMAI Springer Series, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-031-29875-2_5

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