Well-Balanced Path-Consistent Finite Volume EG Schemes for the Two-Layer Shallow Water Equations

  • Michael Dudzinski
  • Mária Lukáčová-Medviďová
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 115)


We present a new path-consistent well-balanced finite volume method within the framework of the evolution Galerkin (FVEG) schemes. The methodology will be illustrated for two layer shallow water equations with source terms modelling the bottom topography and Coriolis forces. The FVEG methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We will derive a suitable path in the phase space that is based on the evolution operator and derive the corresponding path-consistent FVEG scheme. The path-consistent FVEG scheme is well-balanced for the stationary steady states as well as for the steady jets in the rotational frame.


Coriolis Force Bottom Topography Shallow Water Equation Rotational Frame Shallow Water System 
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  1. 1.
    Abgrall, R., Karni, S.: Two–layer shallow water systems: a relaxation approach. SIAM J. Sci. Comput. 31(3), 1603–1627 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bouchut, F., Morales, T.: An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. M2AN 42, 683–698 (2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Castro, M.J., LeFloch, P.G., Munoz-Ruiz, M.L., Parés, C.: Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comp. Phys. 227(17), 8107–8129 (2008)zbMATHCrossRefGoogle Scholar
  5. 5.
    Frings, J.: Well-Balanced Finite Volumes of Higher Order of Accuracy for Two-Layer Shallow Water Flows. Master Thesis, RWTH Aachen (2007)Google Scholar
  6. 6.
    Castro, M.J., López, J.A., Parés, C.: Finite volume simulation of the geostrophic adjustment in a rotating shallow-water system. SIAM J. Sci. Comput. 31, 444–477 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kurganov, A., Petrova, G.: Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31, 1742–1773 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lukáčová-Medvidová, M., Morton, K.W., Warnecke, G.: Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems. SIAM J. Sci. Comput. 26(1), 1–30 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lukáčová-Medvidová, M., Vlk, Z.: Well-balanced Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Source Terms. Int. J. Num. Fluids. 47(10-11), 1165–1171 (2005)CrossRefGoogle Scholar
  10. 10.
    Lukáčová-Medvidová, M., Noelle, S., Kraft, M.: Well-balanced finite volume evolution Galerkin methods for the shallow water equations. J. Comp. Phys. 221, 122–147 (2007)CrossRefGoogle Scholar
  11. 11.
    Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Parés, C., Castro, M.J.: On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. Math. Model. Numer. Anal. 38, 821–852 (2004)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael Dudzinski
    • 1
  • Mária Lukáčová-Medviďová
    • 1
  1. 1.Institute of Numerical SimulationHamburg University of TechnologyHamburgGermany

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