A Discontinuous Galerkin Method for Non-hydrostatic Shallow Water Flows

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 200)

Abstract

In this work a non-hydrostatic depth-averaged shallow water model is discretized using the discontinuous Galerkin (DG) Method. The model contains a non-hydrostatic pressure component, similar to Boussinesq-type equations, which allows for dispersive gravity waves. The scheme is a projection method and consists of a predictor step solving the hydrostatic shallow water equations by the Runge-Kutta DG method. In the correction the non-hydrostatic pressure component is computed by satisfying a divergence constraint for the velocity. This step is discretized by application of the DG discretization to the first order elliptic system. The numerical tests confirm the correct dispersion behavior of the method, and show its validity for simple test cases.

Keywords

Shallow water equations Non-hydrostatic Discontinuous galerkin method 

MSC (2010):

65M08 35Q86 86-08 

References

  1. 1.
    Bai, Y., Cheung, K.F.: Depth-integrated free-surface flow with parameterized non-hydrostatic pressure. Int. J. Numer. Methods Fluids 71(4), 403–421 (2013). doi:10.1002/fld.3664 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Casulli, V., Stelling, G.: Numerical simulation of 3D quasi-hydrostatic, free-surface flows. J. Hydraul. Eng. 124(7), 678–686 (1998)CrossRefGoogle Scholar
  3. 3.
    Cockburn, B.: Discontinuous galerkin methods. Zeitschrift fr Angewandte Mathematik und Mechanik 83(11), 731–754 (2003). doi:10.1002/zamm.200310088 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cui, H., Pietrzak, J., Stelling, G.: Optimal dispersion with minimized poisson equations for non-hydrostatic free surface flows. Ocean Model. 81, 1–12 (2014). doi:10.1016/j.ocemod.2014.06.004 CrossRefGoogle Scholar
  5. 5.
    Dumbser, M., Facchini, M.: A space-time discontinuous Galerkin method for Boussinesq-type equations. Appl. Math. Comput. Part 2 272, 336–346 (2016). doi:10.1016/j.amc.2015.06.052
  6. 6.
    Fringer, O., Gerritsen, M., Street, R.: An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 14(3), 139–173 (2006)CrossRefGoogle Scholar
  7. 7.
    Fuchs, A.: Effiziente parallele Verfahren zur Lösung verteilter, dünnbesetzter Gleichungssysteme eines nichthydrostatischen Tsunamimodells. Ph.D. thesis, AWI, Universität Bremen (2013). http://elib.suub.uni-bremen.de/edocs/00103439-1.pdf
  8. 8.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Publishing Company, Incorporated (2008). doi:10.1007/978-0-387-72067-8
  9. 9.
    Jeschke, A., Pedersen, G.K., Vater, S., Behrens, J.: Depth-averaged non-hydrostatic extension for shallow water equations with quadratic vertical pressure profile: Equivalence to boussinesq-type equations. Int. J. Numer. Methods Fluids (2017). doi:10.1002/fld.4361. http://dx.doi.org/10.1002/fld.4361. (In press)
  10. 10.
    Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M.: Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. J. Fluid Mech. 176, 117–134 (1987). doi:10.1017/S0022112087000594 CrossRefGoogle Scholar
  11. 11.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988). doi:10.1016/0021-9991(88)90177-5 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Stansby, P.K., Zhou, J.G.: Shallow-water flow solver with non-hydrostatic pressure: 2d vertical plane problems. Int. J. Numer. Methods Fluids 28(3), 541–563 (1998). doi:10.1002/(SICI)1097-0363(19980915)28:3<541::AID-FLD738>3.0.CO;2-0 CrossRefMATHGoogle Scholar
  13. 13.
    Stelling, G., Zijlema, M.: An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. Int. J. Numer. Methods Fluids 43(1), 1–23 (2003). doi:10.1002/fld.595 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3 edn. Springer (2009)Google Scholar
  15. 15.
    Ueckermann, M., Lermusiaux, P.: Hybridizable discontinuous Galerkin projection methods for NavierStokes and Boussinesq equations. J. Comput. Phys. 306, 390–421 (2016). doi:10.1016/j.jcp.2015.11.028 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Vater, S., Beisiegel, N., Behrens, J.: A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: one-dimensional case. Adv. Water Resour. 85, 1–13 (2015). doi:10.1016/j.advwatres.2015.08.008 CrossRefGoogle Scholar
  17. 17.
    Walters, R.A.: A semi-implicit finite element model for non-hydrostatic (dispersive) surface waves. Int. J. Numer. Methods Fluids 49(7), 721–737 (2005). doi:10.1002/fld.1019 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversität HamburgHamburgGermany

Personalised recommendations