×

Hydrodynamic instabilities in well-balanced finite volume schemes for frictional shallow water equations. The kinematic wave case. (English) Zbl 1359.76042

Summary: We report the developments of hydrodynamic instabilities in several well-balanced finite volume schemes that are observed during the computation of the temporal evolution of an out-balance flow which is essentially a kinematic wave. The numerical simulations are based on the one-dimensional shallow-water equations for a uniformly sloping bed with hydraulic resistance. Subsequently, we highlight the need of low dissipative high-order well-balanced filter schemes for non-equilibrium flows with variable cut-off wavenumber to compute the out-balance flow under consideration, i.e. the kinematic wave.

MSC:

76B07 Free-surface potential flows for incompressible inviscid fluids
76E99 Hydrodynamic stability
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs

Software:

WENOCLAW
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lighthill, M., Whitham, G.: On kinematic waves. Part I. Flood movement in long rivers. Part II. Theory of traffic flow on long crowded roads. Proc. R. Soc. A (London) 229, 281–345 (1955) · Zbl 0064.20905 · doi:10.1098/rspa.1955.0088
[2] Brock, R.R.: Development of roll-wave trains in open channels. J. Hydraul. Div. 95, 1401–1427 (1969)
[3] Bohorquez, P.: Competition between kinematic and dynamic waves in floods on steep slopes. J. Fluid Mech. 645, 375–409 (2010) · Zbl 1189.76237 · doi:10.1017/S002211200999276X
[4] Castro, M., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math. Comput. 75, 1103–1134 (2006) · Zbl 1096.65082 · doi:10.1090/S0025-5718-06-01851-5
[5] Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007) · Zbl 1120.76046 · doi:10.1016/j.jcp.2007.03.031
[6] Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008) · Zbl 1142.65070 · doi:10.1016/j.jcp.2007.12.005
[7] Wang, W., Yee, H.C., Sjögreen, B., Magin, T., Shu, C.-W.: Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows. J. Comput. Phys. doi: 10.1016/j.jcp.2010.04.033 · Zbl 1343.76037
[8] Burguete, J., García-Navarro, P.: Efficient construction of high-resolution TVD conservative schemes for equations with source terms: application to shallow water flows. Int. J. Numer. Methods Fluids 37(2), 209–248 (2001) · Zbl 1003.76059 · doi:10.1002/fld.175
[9] Črnjarić Žic, N., Vuković, S., Sopta, L.: Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations. J. Comput. Phys. 200, 512–548 (2004) · Zbl 1115.76364 · doi:10.1016/j.jcp.2004.04.012
[10] Needham, D.J., Merkin, J.H.: On roll waves down an open inclined channel. Proc. R. Soc. A (London) 394, 259–278 (1984) · Zbl 0553.76013 · doi:10.1098/rspa.1984.0079
[11] Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009) · Zbl 1203.65135 · doi:10.1007/s10915-008-9239-z
[12] Ketcheson, D.I., LeVeque, R.J.: WENOCLAW: a higher order wave propagation method. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 609–616. Springer, Berlin (2008) · Zbl 1138.65084
[13] Kim, J.W.: High-order compact filters with variable cut-off wavenumber and stable boundary treatment. Comput. Fluids 39, 1168–1182 (2010) · Zbl 1242.76204 · doi:10.1016/j.compfluid.2010.02.007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.