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Bound-Preserving High-Order Schemes for Hyperbolic Equations: Survey and Recent Developments

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Theory, Numerics and Applications of Hyperbolic Problems II (HYP 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 237))

Abstract

Solutions to many hyperbolic equations have convex invariant regions, for example, solutions to scalar conservation laws satisfy the maximum principle, solutions to compressible Euler equations satisfy the positivity-preserving property for density and internal energy. It is, however, a challenge to design schemes whose solutions also honor such invariant regions. This is especially the case for high-order accurate schemes. In this contribution, we survey strategies in the recent literature to design high-order bound-preserving schemes, including a general framework in constructing high-order bound-preserving finite volume and discontinuous Galerkin schemes for scalar and systems of hyperbolic equations through a simple scaling limiter and a convex combination argument based on first-order bound-preserving building blocks, and various flux limiters to design high-order bound-preserving finite difference schemes. We also discuss a few recent developments, including high-order bound-preserving schemes for relativistic hydrodynamics, high-order discontinuous Galerkin Lagrangian schemes, and high-order discontinuous Galerkin methods for radiative transfer equations.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

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  1. Chi-Wang Shu
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Correspondence to Chi-Wang Shu .

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Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

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Shu, CW. (2018). Bound-Preserving High-Order Schemes for Hyperbolic Equations: Survey and Recent Developments. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_44

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