Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T11:52:29.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

Published online by Cambridge University Press:  20 August 2015

Andreas Bollermann ,
Sebastian Noelle  and
Maria Lukáčová-Medvid’ová
Andreas Bollermann*
Affiliation:
IGPM, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany
Sebastian Noelle*
Affiliation:
IGPM, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany
Maria Lukáčová-Medvid’ová*
Affiliation:
Institut fir Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

Keywords

65M0876B1576M1235L5002.60.Cb47.11.Df92.10.SxWell-balanced schemesdry boundariesshallow water equationsevolution Galerkin schemessource terms
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 371 - 404
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alcrudo, F. and García-Navarro, P., A high-resolution Godunov-type scheme in finite volumes for the 2d shallow-water equations, Int. J. Numer. Methods. Fluids., 16(6) (1993), 489–505.Google Scholar
[2] Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25(6) (2004), 2050–2065.Google Scholar
[3] Begnudelli, L. and Sanders, B. F., Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying, J. Hydraul. Eng., 132(4) (2006), 371–384.Google Scholar
[4] Bollermann, A., Lukáčová-Medvid’ová, M. and Noelle, S., Well-balanced finite volume evolution Galerkin methods for the 2d shallow water equations on adaptive grids, In ALGORITMY 2009, 18th Conference on Scientific Computing, pages 81–90, 2009.Google Scholar
[5] Briggs, M., Tsunami benchmark #2: runup of solitary waves on a circular island, http://chl.erdc.usace.army.mil/chl.aspx?p=s&a=Projects; 35.Google Scholar
[6] Briggs, M. J., Synolakis, C. E., Harkins, G. S. and Green, D. R., Laboratory experiments of tsunami runup on a circular island, Pure. Appl. Geophys., 144(3) (1995), 569–593.Google Scholar
[7] Brufau, P. and Garća-Navarro, P., Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique, J. Comput. Phys., 186(2) (2003), 503–526.CrossRefGoogle Scholar
[8] Brufau, P., Vázquez-Cendoón, M. E. and García-Navarro, P., A numerical model for the flooding and drying of irregular domains, Int. J. Numer. Methods. Fluids., 39(3) (2002), 247–275.Google Scholar
[9] Gallardo, J. M., Parés, C. and Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys., 227(1) (2007), 574–601.CrossRefGoogle Scholar
[10] Gallouët, T., H#x00EB;rard, J.-M. and Seguin, N., Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. Fluids., 32(4) (2003), 479–513.Google Scholar
[11] Harten, A. and Hyman, J. M., Self adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys., 50(2) (1983), 235–269.Google Scholar
[12] Hubbard, M.E. and Dodd, N., A 2d numerical model of wave run-up and overtopping, Coast. Eng., 47(1) (2002), 1–26.Google Scholar
[13] Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system, Commun. Math. Sci., 5(1) (2007), 133–160.CrossRefGoogle Scholar
[14] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.CrossRefGoogle Scholar
[15] Liang, Q. and Borthwick, A. G., Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography, Comput. Fluids., 38(2) (2009), 221–234.CrossRefGoogle Scholar
[16] Liang, Q. and Marche, F., Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water. Res., 32(6) (2009), 873–884.Google Scholar
[17] Lukáčová-Medvid’ová, M., Morton, K. W. and Warnecke, G., Evolution Galerkin methods for hyperbolic systems in two space dimensions, Math. Comput., 69(232) (2000), 1355–1384.Google Scholar
[18] Lukáčová-Medvid’ová, M., Morton, K. W. and Warnecke, G., Finite volume evolution Galerkin methods for hyperbolic systems, SIAM J. Sci. Comput., 26(1) (2004), 1–30.Google Scholar
[19] Lukáčová-Medvid’ová, M., Noelle, S. and Kraft, M., Well-balanced finite volume evolution Galerkin methods for the shallow water equations, J. Comput. Phys., 221(1) (2007), 122–147.Google Scholar
[20] Lukáčová-Medvid’ová, M., Saibertova, J. and Warnecke, G., Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comput. Phys., 183(30) (2002), 533–562.Google Scholar
[21] Lukáčová-Medvid’ová, M. and Tadmor, E., On the entropy stability of the Roe-type finite volume methods, In Tadmor, E., Liu, J.-G., and Tzavaras, A. E., editors, Hyperbolic Problems: Theory, Numerics and Applications, Volume 67 of Proceedings of Symposia in Applied Mathematics, pages 765–774, 2009.Google Scholar
[22] Lukáčová-Medvid’ová, M., Warnecke, G. and Zahaykah, Y., On the boundary conditions for eg methods applied to the two-dimensional wave equation system, ZAMM., 84(4) (2004), 237–251.Google Scholar
[23] Marche, F., Bonneton, P., Fabrie, P. and Seguin, N., Evaluation of well-balanced bore-capturing schemes for 2d wetting and drying processes, Int. J. Numer. Methods. Fluids., 53(5) (2007), 867–894.Google Scholar
[24] Marquina, A., Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws, SIAM J. Sci. Comput., 15(4) (1994), 892–915.Google Scholar
[25] Ricchiuto, M. and Bollermann, A., Accuracy of stabilized residual distribution for shallow water flows including dry beds, In Tadmor, E., Liu, J.-G., and Tzavaras, A. E., editors, Hyperbolic Problems: Theory, Numerics and Applications, Volume 67 of Proceedings of Symposia in Applied Mathematics, pages 889–898, 2009.Google Scholar
[26] Ricchiuto, M. and Bollermann, A., Stabilized residual distribution for shallow water simulations, J. Comput. Phys., 228(4) (2009), 1071–1115.Google Scholar
[27] Roe, P. L., Sonic flux formulae, SIAM J. Sci. Stat. Comput., 13(2) (1992), 611–630.Google Scholar
[28] Seaïd, M., Non-oscillatory relaxation methods for the shallow-water equations in one and two space dimensions, Int. J. Numer. Methods. Fluids., 46(5) (2004), 457–484.Google Scholar
[29] Synolakis, C. E., The Runup of Long Waves, PhD thesis, California Institute of Technology, (1986).Google Scholar
[30] Synolakis, C. E., The runup of solitary waves, J. Fluid. Mech., 185 (1987), 523–545.CrossRefGoogle Scholar
[31] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta. Numerica., 12(-1) (2003), 451–512.Google Scholar
[32] Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid. Mech., 107 (1981), 499–508.Google Scholar