×

Existence result for the coupling of shallow water and Borda-Carnot equations with Riemann data. (English) Zbl 1447.35266

Summary: We consider a subcritical flow in a sudden expansion canal. The flow is given by 1D Saint-Venant equations on each side of the expansion together with mass conservation and Borda-Carnot conditions at the expansion. We show, following the ideas of our work [ibid. 10, No. 3, 431–460 (2013; Zbl 1277.35239)], that the linear Riemann problem always has a unique solution while for the nonlinear problem, a unique solution is obtained only under a certain condition.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
15A18 Eigenvalues, singular values, and eigenvectors
35L50 Initial-boundary value problems for first-order hyperbolic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Citations:

Zbl 1277.35239

Software:

HE-E1GODF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambroso, A., Chalons, C., Coquel, F., Godlewski, E., Lagoutire, F., Raviart, P.-A. and Seguin, N., Coupling of multiphase flow models, 11th Int. Topical Meeting on Nuclear Thermal-Hydraulics, October 2005, Avignon, France, .
[2] Banda, M. K., Herty, M. and Klar, A., Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media1(2) (2006) 295-314. · Zbl 1109.76052
[3] Bernetti, R., Titarev, V. A. and Toro, E. F., Exact solution of the Riemann problem for shallow water equations with discontinuous bottom geometry, J. Comput. Phys.227 (2008) 3212-3243. · Zbl 1132.76027
[4] Goudiaby, M. S. and Kreiss, G., A Riemann problem at a junction of open canals, J. Hyp. Diff. Eq.10(3) (2013) 431-460. · Zbl 1277.35239
[5] Briani, M., Piccoli, B. and Qiu, J. M., Notes on RKDG Methods for Shallow-Water equations in canal networks, J. Sci. Comput.68(3) (2016) 1101-1123. · Zbl 1437.76020
[6] Colombo, R. M and Garavello, M., A well posed Riemann problem for the \(p\)-system at a junction, Netw. Heterog. Media1(3) (2006) 495-511. · Zbl 1116.35086
[7] Colombo, R. M, Goatin, P. and Rosini, M. D., On the modeling and management of traffic, ESAIM: Math. Model. Numer. Anal.45(5) (2011) 853-872. · Zbl 1267.90032
[8] Cunge, J. A., Holly, F. M. and Verwey, A., Practical Aspects of Computational River Hydraulics (Pitman, Boston, 1980).
[9] De Halleux, J., Prieur, C., Coron, J.-M., d’Andra-Novel, B. and Bastin, G., Boundary feedback control in networks of open channels, Automatica39(8) (2003) 1365-1376. · Zbl 1175.93108
[10] De Saint-Venant, B., Théorie du mouvement non-permanent des eaux avec application aux crues des rivières et à l’introduction des marées dans leur lit, Compt. Rendus Acad. Sci.73 (1871) 148-154, 237-240. · JFM 03.0482.04
[11] Elshobaki, M., Valiani, A. and Caleffi, V., Junction Riemann problem for one-dimensional shallow water equations with bottom discontinuities and channels width variations, J. Hyp. Diff. Eq.15(2) (2018) 191-217. · Zbl 1398.35165
[12] Garavello, M. and Piccoli, B., Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations31(1-3) (2006) 243-275. · Zbl 1090.90032
[13] Godlewski, E. and Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case, Numer. Math.97(1) (2004) 81-130. · Zbl 1063.65080
[14] Boutin, B., Chalons, C. and Raviart, P.-A, Existence result for the coupling problem of two scalar conservation laws with Riemann initial data, Math. Models Methods Appl. Sci.10(10) (2010) 1859-1898. · Zbl 1211.35017
[15] Gugat, M., Leugering, G. and Georg Schmidt, E. J. P., Global controllability between steady supercritical flows in channel networks, Math. Meth. Appl. Sci.27 (2004) 781-802. · Zbl 1047.93028
[16] Herty, M. and Rascle, M., Coupling conditions for a class of “second-order” models for traffic flow, SIAM J. Math. Anal.38(2) (2006) 595-616. · Zbl 1296.35094
[17] Holden, H. and Risebro, N. H., Riemann problem with a kink, SIAM. J. Math. Anal.30 (1999) 497-515. · Zbl 0926.35087
[18] Holden, H., On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math.40(2) (1987) 229-264. · Zbl 0637.35051
[19] C. E. Kindsvater, Energy losses associated with abrupt enlargements in pipes, http://pubs.usgs.gov/wsp/1369b/report.pdf.
[20] LeFloch, P. G. and Thanh, M. D., The Riemann problem for the shallow water equations with discontinuous topography, Comm. Math. Sci.5 (2007) 865-885. · Zbl 1145.35082
[21] Leugering, G. and Georg Schmidt, E. J. P., On the modelling and stabilization of flows in networks of open canal, SIAM J. Control Optim.41(1) (2002) 164-180. · Zbl 1024.76009
[22] Leveque, R. J., Finite Volume Methods for Hyperbolic Problems, , (Cambridge, USA, 2002). · Zbl 1010.65040
[23] Li, T., Exact boundary controlability of unsteady flows in a network of open canals, Math. Nachr.278(3) (2005) 278-289. · Zbl 1066.93005
[24] Marchesin, D. and Paes-Leme, P. J., A Riemann problem in gas dynamics with bifurcation, Comp. Maths. Appl.12A(15) (1986) 433-455. · Zbl 0611.35060
[25] Marigo, A., Entropic solutions for irrigation networks, SIAM J. Appl. Math.70(5) (2010) 1711-1735. · Zbl 1201.35125
[26] Mercier, M., Traffic flow modelling with junctions, J. Math. Anal. Appl.350(1) (2009) 369-383. · Zbl 1162.90006
[27] Qilong, G. and Li, T., Exact boundary observability of unsteady flows in a tree-like network of open canals, Math. Meth. Appl. Sci.31(4) (2008) 395-418. · Zbl 1155.35401
[28] Reigstad, G. A., Flatten, T., Haugen, N. E. and Ytrehus, T., Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyp. Diff. Eq.12(1) (2015) 37-59. · Zbl 1321.35148
[29] Reigstad, G. A., Numerical network models and entropy principles for isothermal junction flow, Netw. Heterog. Media9(1) (2014) 65-95. · Zbl 1304.35523
[30] Rosatti, G. and Begnudelli, L., The Riemann problem for the one-dimensional, free-surface shallow water equations with a bed step: Theoretical analysis and numerical simulations, J. Comput. Phys.229 (2010) 760-787. · Zbl 1253.76014
[31] Toro, E. F., Riemann problem and the WAF method for solving the two-dimensional shallow water equations with a bed step: Theoretical analysis and numerical simulations, Phil. Trans. R. Soc. Lond., A238 (1992) 43-68. · Zbl 0747.76027
[32] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer-Verlag, Berlin, 1999). · Zbl 0923.76004
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.