×

zbMATH — the first resource for mathematics

A numerical technique for applying time splitting methods in shallow water equations. (English) Zbl 1410.76248
Summary: In this paper, we analyze the use of time splitting techniques for solving shallow water equations. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This work shows that these schemes must be well balanced in the meaning expressed by J. M. Greenberg and A.-Y. Le Roux [SIAM J. Numer. Anal. 33, No. 1, 1–16 (1996; Zbl 0876.65064)]. More specifically, we analyze in what cases it is enough to verify an approximate C-property and in which cases it is required to verify an Exact C-property (see [A. Bermúdez et al., Comput. Methods Appl. Mech. Eng. 155, No. 1–2, 49–72 (1998; Zbl 0961.76047); A. Bermúdez and M. E. Vázquez, Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)]). We also discuss this technique in two dimensions and include some numerical tests in order to justify our argument.
MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bermúdez, A.; Dervieux, A.; Desideri, J.; Vázquez, M., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput Methods Appl Mech Eng, 155, 49-72, (1998) · Zbl 0961.76047
[2] Bermúdez, A.; Vázquez, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comp Fluids, 23, 1049-1071, (1994) · Zbl 0816.76052
[3] Berthon, C.; Marche, F.; Turpault, R., An efficient scheme on wet/dry transitions for shallow water equations with friction comp, Fluids, 48, 192-201, (2011) · Zbl 1271.76178
[4] Bollermann, A.; Chen, G.; Kurganov, A.; Noelle, S., A well-balanced reconstruction of wet/dry fronts for the shallow water equations, J Sci Comput, 56, 267-290, (2013) · Zbl 1426.76337
[5] Brufau, P.; Vázquez-Cendón, M.; García-Navarro, P., A numerical model for the flooding and drying of irregular domain, Int J Numer Methods Fluids, 39, 247-275, (2002) · Zbl 1094.76538
[6] Castro, M.; Ferreiro Ferreiro, A.; García-Rodríguez, J.; González-Vida, J.; Macías, J.; Parés, C., The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems, Math Comput Model, 42, 419-439, (2005) · Zbl 1121.76008
[7] Greenberg, J.; Leroux, A., A well-balanced scheme for the numerical processing of source terms in hyperbolic, SINUM, 33, 1-16, (1996) · Zbl 0876.65064
[8] Holdahl, R.; Holden, H.; Lie, K.-A., Unconditionally stable splitting methods for the shallow water equations, BIT, 39, 451-472, (1999) · Zbl 0945.76059
[9] Holden, H.; Karlsen, K.; Lie, K.-A.; Risebro, N., Splitting methods for partial differential equations with rough solutions. Analysis and MATLAB programs, (2010), European Mathematical Society Publishing · Zbl 1191.35005
[10] Holden, H.; Karlsen, K. H.; Risebro, N.; Tao, T., Operator splitting for the KDV equation, Math Comput, 80, 821-846, (2011) · Zbl 1219.35235
[11] Horváth, Z.; Waser, J.; Perdigão, R.; Konev, A.; Blöschl, G., A two-dimensional numerical scheme of dry/wet fronts for the Saint-Venant system of shallow water equations int, J Numer Meth Fluids, 77, 159-182, (2015)
[12] Huang, Y.; Zhang, N.; Pei, Y., Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography, Eng Appl Comput Fluid Mech, 7, 40-54, (2013)
[13] LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J Comput Phys, 146, 346-365, (1998) · Zbl 0931.76059
[14] LeVeque, R.; Yee, H., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J Comput Phys, 86, 187-210, (1990) · Zbl 0682.76053
[15] Lubich, C., On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math Comput, 77, 2141-2153, (2008) · Zbl 1198.65186
[16] Ma, D.-J.; Sun, D.-J.; Yin, X.-Y., Solution of the 2-D shallow water equations with source terms in surface elevation splitting form, Int J Numer Meth Fluids, 55, 431-454, (2007) · Zbl 1388.76143
[17] Martínez V. To split or not to split, that is the question in some shallow water equations. 2012. ArXiv:1211.6655v1 [math.NA], 28 Nov.
[18] Skiba, Y.; Filatov, D., Conservative arbitrary order finite difference schemes for shallow-water flows, J Comput Appl Math, 218, 579-591, (2008) · Zbl 1225.76216
[19] Toro E.. Riemann solvers and numerical methods for fluid dynamics. 2009. Third Edition, Springer. · Zbl 1227.76006
[20] Vázquez-Cendón, M., Improved treatment of source terms in upwind 389 schemes for the shallow water equations in channels with irregular geometry, J Comput Phys, 148, 497-526, (1999) · Zbl 0931.76055
[21] Wicker, L.; Skamarock, W., Time-splitting methods for elastic models using forward time schemes, Mon Wea Rev, 130, 2088-2097, (2002)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.