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.
Mots-clés : hyperbolic systems of conservation and balance laws, semi-discrete schemes, Saint-Venant system of shallow water equations, non-oscillatory reconstructions, channels with irregular geometry
@article{M2AN_2014__48_3_665_0,
author = {Balb\'as, Jorge and Hernandez-Duenas, Gerardo},
title = {A positivity preserving central scheme for shallow water flows in channels with wet-dry states},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {665--696},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013106},
mrnumber = {3177861},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013106/}
}
TY - JOUR AU - Balbás, Jorge AU - Hernandez-Duenas, Gerardo TI - A positivity preserving central scheme for shallow water flows in channels with wet-dry states JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 665 EP - 696 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013106/ DO - 10.1051/m2an/2013106 LA - en ID - M2AN_2014__48_3_665_0 ER -
%0 Journal Article %A Balbás, Jorge %A Hernandez-Duenas, Gerardo %T A positivity preserving central scheme for shallow water flows in channels with wet-dry states %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 665-696 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013106/ %R 10.1051/m2an/2013106 %G en %F M2AN_2014__48_3_665_0
Balbás, Jorge; Hernandez-Duenas, Gerardo. A positivity preserving central scheme for shallow water flows in channels with wet-dry states. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 665-696. doi : 10.1051/m2an/2013106. http://archive.numdam.org/articles/10.1051/m2an/2013106/
[1] and , Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31 (2009) 1603-1627. | MR | Zbl
[2] , The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163 (1986) 27-58. | MR
[3] and , Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 186 (1986) 27-51. | MR | Zbl
[4] , , , and , A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065. | MR | Zbl
[5] and , A central scheme for shallow water flows along channels with irregular geometry. ESAIM: M2AN 43 (2009) 333-351. | Numdam | MR | Zbl
[6] , , and , A well-balanced reconstruction of wet/dry fronts for the shallow water equations. J. Sci. Comput. (2011) 1-24.
[7] , and , Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10 (2011) 371-404. | MR | Zbl
[8] , Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004). | MR | Zbl
[9] , and , 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. | Numdam | MR | Zbl
[10] , , , , and , Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202-235. | MR | Zbl
[11] , and , Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Models Methods Appl. Sci. 17 (2007) 2055-2113. | MR | Zbl
[12] , and , 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. | MR | Zbl
[13] and , 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. | Zbl
[14] and , On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 951-979. | Zbl
[15] , Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 227 (2008) 3089-3113. | MR
[16] , and , Strong stability-preserving high-order time discretization methods. SIAM Review 43 (2001) 89-112. | MR | Zbl
[17] and , Shallow water flows in channels. J. Sci. Comput. 48 (2011) 190-208.
[18] , A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN (2001) 35 631-645. | Numdam | MR | Zbl
[19] and , 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.
[20] and , Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | MR | Zbl
[21] and , A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5 (2007) 133-160. | MR | Zbl
[22] and , Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31 (2009) 1742-1773. | MR | Zbl
[23] and , New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | MR | Zbl
[24] , Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365. | MR | Zbl
[25] , , and , Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499. | MR | Zbl
[26] , and , High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29-58. | MR | Zbl
[27] and , A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl
[28] , 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. | Zbl
[29] , 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.
[30] , 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. | Zbl
[31] . 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. | Zbl
[32] and , 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. | Zbl
[33] , and , On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48 (2011) 339-349.
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