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

Balbás, Jorge; Hernandez-Duenas, Gerardo

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 665-696.

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.

Classification : 76M12, 35L65
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

[1] R. Abgrall and S. Karni, Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31 (2009) 1603-1627. | MR | Zbl

[2] L. Armi, The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163 (1986) 27-58. | MR

[3] L. Armi and D.M. Farmer, Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 186 (1986) 27-51. | MR | Zbl

[4] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, 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] J. Balbás and S. Karni, A central scheme for shallow water flows along channels with irregular geometry. ESAIM: M2AN 43 (2009) 333-351. | Numdam | MR | Zbl

[6] 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.

[7] A. Bollermann, S. Noelle and M. Lukáčová-Medvidóvá, Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10 (2011) 371-404. | MR | Zbl

[8] 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). | MR | Zbl

[9] M. Castro, J. Macías and C. Parés, 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] M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202-235. | MR | Zbl

[11] M.J. Castro, A. Pardo Milanés and C. Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Models Methods Appl. Sci. 17 (2007) 2055-2113. | MR | Zbl

[12] N. Črnjarić-Žic, S. Vuković and L. Sopta, 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] D.M. Farmer and L. Armi, 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] P. Garcia-Navarro and M.E. Vazquez-Cendon, On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 951-979. | Zbl

[15] D.L. George, 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] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Review 43 (2001) 89-112. | MR | Zbl

[17] G. Hernández-Dueñas and S. Karni, Shallow water flows in channels. J. Sci. Comput. 48 (2011) 190-208.

[18] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN (2001) 35 631-645. | Numdam | MR | Zbl

[19] 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.

[20] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | MR | Zbl

[21] A. Kurganov and G. Petrova, 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] A. Kurganov and G. Petrova, Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31 (2009) 1742-1773. | MR | Zbl

[23] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | MR | Zbl

[24] R.J. Leveque, 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] S. Noelle, N. Pankratz, G. Puppo and J.R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499. | MR | Zbl

[26] S. Noelle, Y. Xing and C.-W. Shu, 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] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl

[28] 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. | Zbl

[29] 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.

[30] B. Van Leer, 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] M. E. Vázquez-Cendón. 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] 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. | Zbl

[33] Yulong Xing, Chi-Wang Shu and Sebastian Noelle, On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48 (2011) 339-349.