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A Review of Cartesian Grid Active Flux Methods for Hyperbolic Conservation Laws

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Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems (FVCA 2023)

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Abstract

In 2011, Eymann and Roe introduced a new class of truly multidimensional finite volume methods. These so-called Active Flux methods use concepts, which are quite different from concepts that are typically used in numerical schemes for hyperbolic conservation laws. In particular the method is based on the use of point values as well as cell average values, it uses a continuous reconstruction and does not rely on the use of Riemann solvers. We will review the current state of the art of Cartesian grid Active Flux methods and present recent results of our group. These include a discussion of different evolution formulas for the update of the point values, results on the linear stability of the resulting methods as well as a discussion of limiters. Furthermore, we explore the use of Active Flux methods on Cartesian grids with adaptive mesh refinement.

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Authors and Affiliations

  1. Institute of Mathematics, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany

    Erik Chudzik & Christiane Helzel

Authors
  1. Erik Chudzik
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  2. Christiane Helzel
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Correspondence to Christiane Helzel .

Editor information

Editors and Affiliations

  1. IRMA, Strasbourg, France

    Emmanuel Franck

  2. Weierstrass Institute, Berlin, Germany

    Jürgen Fuhrmann

  3. IRMA, Strasbourg, France

    Victor Michel-Dansac

  4. IRMA, Strasbourg, France

    Laurent Navoret

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Chudzik, E., Helzel, C. (2023). A Review of Cartesian Grid Active Flux Methods for Hyperbolic Conservation Laws. In: Franck, E., Fuhrmann, J., Michel-Dansac, V., Navoret, L. (eds) Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems. FVCA 2023. Springer Proceedings in Mathematics & Statistics, vol 432. Springer, Cham. https://doi.org/10.1007/978-3-031-40864-9_6

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