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Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

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Abstract

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the L 1-stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.

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References

  1. M. Brio, A. R. Zakharian, and G. M. Webb: Two-dimensional Riemann solver for Euler equations of gas dynamics. J. Comput. Phys. 167 (2001), 177–195.

    Article  MATH  Google Scholar 

  2. M. Fey: Multidimensional upwinding, Part II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998), 181–199.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Kröger, S. Noelle: Numerical comparison of the Method of Transport to a standard scheme. Comput. Fluids 34 (2003), 541–560.

    Article  Google Scholar 

  4. A. Kurganov, S. Noelle, and G. Petrova: Semidiscrete central-unpwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001), 707–740.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. J. LeVeque: Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Phys. 131 (1997), 327–353.

    Article  MATH  Google Scholar 

  6. J. Li, M. Lukáčová-Medviďová, G. Warnecke: Evolution Galerkin schemes for the two-dimensional Riemann problems for the wave equation systems. Discrete Contin. Dyn. Syst. (A) 9 (2003), 559–576.

    Article  MATH  Google Scholar 

  7. M. Lukáčová-Medviďová, K. W. Morton, and G. Warnecke: Evolution Galerkin methods for hyperbolic systems in two space dimensions. Math. Comput. 69 (2000), 1355–1384.

    Article  Google Scholar 

  8. M. Lukáčová-Medviďová, K. W. Morton, and G. Warnecke: Finite volume evolution Galerkin methods for Euler equations of gas dynamics. Internat. J. Numer. Methods Fluids 40 (2002), 425–434.

    Article  MathSciNet  Google Scholar 

  9. M. Lukáčová-Medviďová, K. W. Morton, and G. Warnecke: Finite volume evolution Galerkin (FVEG) methods for hyperbolic systems. SIAM J. Sci. Comput. 26 (2004), 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Lukáčová-Medviďová, J. Saibertová, and G. Warnecke: Finite volume evolution Galerkin methods for nonlinear hyperbolic systems. J. Comput. Phys. 183 (2002), 533–562.

    Article  MathSciNet  Google Scholar 

  11. M. Lukáčová-Medviďová, G. Warnecke: Lax-Wendroff type second order evolution Galerkin methods for multidimensional hyperbolic systems. East-West Numer. Math. 8 (2000), 127–152.

    Google Scholar 

  12. M. Lukáčová-Medviďová, G. Warnecke, and Y. Zahaykah: On the stability of the evolution Galerkin schemes applied to a two-dimensional wave equation system. SIAM J. Numer. Anal. (2006).

  13. K. W. Morton: On the analysis of finite volume methods for evolutionary problems. SIAM J. Numer. Anal. 35 (1998), 2195–2222.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Noelle: The MOT-ICE: a new high-resolution wave-propagation algorithm for multi-dimensional systems of conservative laws based on Fey’s method of transport. J. Comput. Phys. 164 (2000), 283–334.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Ostkamp: Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems. Math. Methods Appl. Sci. 20 (1997), 1111–1125.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Ostkamp: Multidimensional characteristic Galerkin schemes and evolution operators for hyperbolic systems. PhD. thesis. University Hannover, 1995.

  17. J. Saibertová: Genuinely multidimensional finite volume schemes for systems of conservation laws. PhD. thesis. Technical University Brno, 2003.

  18. Y. Zahaykah: Evolution Galerkin schemes and discrete boundary conditions for multidimensional first order systems. PhD. thesis. University of Magdeburg, 2002.

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Authors and Affiliations

  1. Institute of Numerical Simulation, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg, Germany

    Mária Lukáčová-Medviďová

  2. Institute of Mathematics, University of Technology Brno, Technická 2, CZ-616 00, Brno, Czech Republic

    Jitka Saibertová-Zatočilová

Authors
  1. Mária Lukáčová-Medviďová
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  2. Jitka Saibertová-Zatočilová
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Additional information

This research has been supported under the VW-Stiftung grant I 76 859, by the grant No 201/03 0570 of the Grant Agency of the Czech Republic, by the Deutsche Forschungsgemeinschaft grant GK 431 and partially by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282.

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Lukáčová-Medviďová, M., Saibertová-Zatočilová, J. Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics. Appl Math 51, 205–228 (2006). https://doi.org/10.1007/s10492-006-0012-z

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  • DOI: https://doi.org/10.1007/s10492-006-0012-z

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