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A central scheme for shallow water flows along channels with irregular geometry

  • Jorge Balbás (a1) and Smadar Karni (a2)

Abstract

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm.

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[1] R. Abgrall and S. Karni, A relaxation scheme for the two layer shallow water system, in Proceedings of the 11th International Conference on Hyperbolic Problems (Lyon, 2006), Springer (2008) 135–144.
[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 (2004) 20502065
[3] J. Balbás and E. Tadmor, Nonoscillatory central schemes for one- and two-dimensional magnetohydrodynamics equations. ii: High-order semidiscrete schemes. SIAM J. Sci. Comput. 28 (2006) 533–560.
[4] Bermudez, A. and Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 10491071.
[5] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel, Switzerland, Berlin (2004).
[6] M.J. Castro, J. Macias and C. Pares, A Q-scheme for a class of systems of coupled conservation laws with source terms. Application to a two-layer 1-d shallow water system. ESAIM: M2AN 35 (2001) 107–127.
[7] 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) 202235.
[8] 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) 512548.
[9] Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89112.
[10] Greenberg, J.M. and Le Roux, A.Y., Well-balanced scheme for the processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 116.
[11] Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357393.
[12] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms. ESAIM: M2AN 35 (2001) 631645.
[13] Kurganov, A. and Levy, D., Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397425.
[14] 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) 133160.
[15] Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241282.
[16] Kurganov, A., Noelle, S. and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707740.
[17] LeVeque, R.J., Balancing source terms and flux gradients in high resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346365.
[18] Nessyahu, H. and Tadmor, E., Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408463.
[19] 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) 474499.
[20] 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) 2958.
[21] C. Pares and M. Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821–852.
[22] Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201231.
[23] G. Russo, Central schemes for balance laws, in Hyperbolic problems: theory, numerics, applications, Vols. I, II (Magdeburg, 2000), Internat. Ser. Numer. Math. 140, Birkhäuser, Basel (2001) 821–829.
[24] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. Comput. Phys. 83 (1989) 32–78.
[25] Thacker, W.C., Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics Digital Archive 107 (1981) 499508.
[26] B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J. Comput. Phys. 135 (1997) 229–248.
[27] 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) 497526.
[28] 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.

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A central scheme for shallow water flows along channels with irregular geometry

  • Jorge Balbás (a1) and Smadar Karni (a2)

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