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Abstract

We present recent work in progress towards the development of a third order accurate Cartesian grid cut cell method for the approximation of hyperbolic conservation laws in complex geometries. Our cut cell method is based on the Active Flux method of Eymann and Roe, a new finite volume method, which evolves both cell average values and point values of the conserved quantities. The evolution of the point values leads to an automatic stabilisation of the cut cell update, i.e. the method is stable for time steps that are appropriate for the regular cells. While most of the existing cut cell stabilisation methods lead to a loss of accuracy, we show that it is possible to obtain third order accurate results. In this contribution we restrict our considerations to the linear transport equation in one and two space dimensions.

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References

  1. Barsukow, W., Hohm, J., Klingenberg, C., Roe, P.L.: The Active Flux scheme on cartesian grids and its low mach number limit. J. Sci. Comput. 81, 594–622 (2019)

    Article  MathSciNet  Google Scholar 

  2. Engwer, C., May, S., Nüssing, C., Streitbürger, F., A stabilized discontinuous Galerkin cut cell method for discretizing the linear transport equation. (2019). arXiv:1906.05642

  3. Eymann, T.A., Roe, P.: Active Flux schemes. In: 49th AIAA Aerospace Science meeting (2011)

    Google Scholar 

  4. Eymann, T.A., Roe, P.L.: Multidimensional Active Flux schemes. In: AIAA Conference Paper (2013)

    Google Scholar 

  5. Helzel, C., Kerkmann, D., Scandurra, L.: A new ADER method inspired by the Active Flux method. J. Sci. Comput. 80, 1463–1497 (2019)

    Article  MathSciNet  Google Scholar 

  6. van Drosselaer, J.L.M., Kraaijevanger, J.F.B.M., Spijker, M.N.: Linear stability analysis in the numerical solution of initial value problems. Acta Numer. 2, 199–237 (1993)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the DFG through HE 4858/4-1.

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Correspondence to David Kerkmann .

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Helzel, C., Kerkmann, D. (2020). An Active Flux Method for Cut Cell Grids. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_47

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