Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T13:11:34.524Z Has data issue: false hasContentIssue false
  • English
  • Français

A positivity preserving central scheme for shallow water flows in channels with wet-dry states∗

Published online by Cambridge University Press:  20 January 2014

Jorge Balbás  and
Gerardo Hernandez-Duenas
Jorge Balbás
Affiliation:
Department of Mathematics. California State University, Northridge. 18111 Nordhoff St. Northridge, CA 91330-8313, USA.. [email protected]
Gerardo Hernandez-Duenas
Affiliation:
Department of Mathematics, University of Wisconsin – Madison. 480 Lincoln Dr., Madison, Wi 53706-1325, USA.; [email protected]
Get access

Abstract

We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Keywords

Hyperbolic systems of conservation and balance lawssemi-discrete schemesSaint-Venant system of Shallow Water equationsnon-oscillatory reconstructionschannels with irregular geometry
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 665 - 696
Copyright
© EDP Sciences, SMAI, 2014

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

Abgrall, R. and Karni, S., Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31 (2009) 1603–1627. Google Scholar
Armi, L., The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163 (1986) 27–58. Google Scholar
Armi, L. and Farmer, D.M., Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 186 (1986) 27–51. Google Scholar
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 (2004) 2050–2065. Google Scholar
Balbás, J. and Karni, S., A central scheme for shallow water flows along channels with irregular geometry. ESAIM: M2AN 43 (2009) 333–351. Google Scholar
A. Bollermann, G. Chen, A. Kurganov and S. Noelle, A well-balanced reconstruction of wet/dry fronts for the shallow water equations. J. Sci. Comput. (2011) 1–24.
Bollermann, A., Noelle, S. and Lukáčová-Medvidóvá, M., Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10 (2011) 371–404. Google Scholar
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).
Castro, M., Macías, J. and Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107–127. Google Scholar
Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C. and Vázquez-Cendón, M.E., Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202–235. Google Scholar
Castro, M.J., Pardo Milanés, A. and Parés, C., Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Models Methods Appl. Sci. 17 (2007) 2055–2113. Google Scholar
Črnjarić-Žic, N., Vuković, S. and Sopta, L., Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. J. Comput. Phys. 200 (2004) 512–548. Google Scholar
Farmer, D.M. and Armi, L., Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow. J. Fluid Mech. 164 (1986) 53–76. Google Scholar
Garcia-Navarro, P. and Vazquez-Cendon, M.E., On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 951–979. Google Scholar
George, D.L., Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 227 (2008) 3089–3113. Google Scholar
Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods. SIAM Review 43 (2001) 89–112. Google Scholar
Hernández-Dueñas, G. and Karni, S., Shallow water flows in channels. J. Sci. Comput. 48 (2011) 190–208. Google Scholar
Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN (2001) 35 631–645. Google Scholar
S. Karni and G. Hernández-Dueñas, A scheme for the shallow water flow with area variation. AIP Conference Proceedings. Vol. 1168 of International Conference Numer. Anal. Appl. Math., Rethymno, Crete, Greece. American Institute of Physics (2009) 1433–1436.
Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397–425. Google Scholar
Kurganov, A. and Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133–160. Google Scholar
Kurganov, A. and Petrova, G., Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31 (2009) 1742–1773. Google Scholar
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241–282. Google Scholar
LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346–365. Google Scholar
Noelle, S., Pankratz, N., Puppo, G. and Natvig, J.R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474–499. Google Scholar
Noelle, S., Xing, Y. and Shu, C.-W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29–58. Google Scholar
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. Google Scholar
P.L. Roe, Upwind differencing schemes for hyperbolic conservation laws with source terms. Nonlinear hyperbolic problems (St. Etienne, 1986). In vol. 1270 of Lecture Notes in Math. Springer, Berlin (1987) 41–51.
G. Russo, Central schemes for balance laws. Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000). In vol. 140 of Internat. Ser. Numer. Math. Birkhäuser, Basel (2001) 821–829.
van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32 (1979) 101–136; J. Comput. Phys. 135 (1997) 227–248. Google Scholar
Vázquez-Cendón, M. E.. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148 (1999) 497–526. Google Scholar
S. Vuković and L. Sopta, High-order ENO and WENO schemes with flux gradient and source term balancing. In Applied mathematics and scientific computing (Dubrovnik, 2001). Kluwer/Plenum, New York (2003) 333–346.
Xing, Yulong, Shu, Chi-Wang and Noelle, Sebastian, On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48 (2011) 339–349. Google Scholar