Hostname: page-component-7b7567c8c-jswmv Total loading time: 0 Render date: 2026-11-15T10:23:05.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Shock Profiles for the Shallow-Water Exner Models

Published online by Cambridge University Press:  28 May 2015

C. Berthon,
B. Boutin and
R. Turpault
C. Berthon
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière 44322 Nantes, France
B. Boutin
Affiliation:
Université de Rennes 1, Institut de Recherche Mathématiques de Rennes, Rennes, France
R. Turpault*
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière 44322 Nantes, FranceBordeaux-INP, Institut de Mathématiques de Bordeaux, 351 Cours de la Libération 33400 Talence, France
*
*Corresponding author. Email: rodolphe.turpault@univ-nantes.fr (R. Turpault)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This article is devoted to analyze some ambiguities coming from a class of sediment transport models. The models under consideration are governed by the coupling between the shallow-water and the Exner equations. Since the PDE system turns out to be an hyperbolic system in non conservative form, ambiguities may occur as soon as the solution contains shock waves. To enforce a unique definition of the discontinuous solutions, we adopt the path-theory introduced by Dal Maso, LeFLoch and Murat [18]. According to the path choices, we exhibit several shock definitions and we prove that a shock with a constant propagation speed and a given left state may connect an arbitrary right state. As a consequence, additional assumptions (coming from physical considerations or other arguments) must be chosen to enforce a unique definition. Moreover, we show that numerical ambiguities may still exist even when a path is chosen to select the system's solution.

Keywords

35L6776N9965M08Exner modelsnonconservative hyperbolic systemsshock wave solutionsgeneralized Rankine-Hugoniot relationshugoniot curvesdam-break simulations

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 3 , June 2015, pp. 267 - 294
Copyright
Copyright © Global-Science Press 2015 

References

[1]Abgrall, R. and Karni, S., A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), pp. 2759–2763.CrossRefGoogle Scholar
[2]Audusse, E., Berthon, C., Chalons, C., Delestre, O., Goutal, N., Jodeau, M., Sainte-Marie, J., Giesselmann, J. and Sadaka, G., Sediment transport modelling: relaxation schemes for Saint-Venant-Exner and three layer models, ESAIM: Proceedings, 38 (2012), pp. 78–98.CrossRefGoogle Scholar
[3]Berthon, C., Coquel, F. and Lefloch, P.G., Why many theories of shock waves are necessary: kinetic relations for non-conservative systems, Proc. Roy. Soc. Edinburgh, 142(01) (2012), pp. 137.CrossRefGoogle Scholar
[4]Berthon, C. and Coquel, F., Nonlinear projection methods for multi-entropies Navier-Stokes systems, Math. Comput., 76 (2007), pp. 1163–1194.Google Scholar
[5]Berthon, C. and Coquel, F., Shock layers for turbulence models, Math. Models Methods Appl. Sci., 18 (2008), pp. 1443–1479.Google Scholar
[6]Berthon, C. and Nkonga, B., Multifluid numerical approximations based on a multi-pressure formulation, Comput. Fluids, 36 (2007), pp. 467–479.Google Scholar
[7]Castro Díaz, M.J., Fernández-Nieto, E.D. and Ferreiro, A.M., Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods, Comput. Fluids, 37 (2008), pp. 299–316.Google Scholar
[8]Castro Díaz, M.J., Fernández-Nieto, E.D., Ferreiro, A.M. and Parès, C., Two-dimensional sediment transport models in shallow water equations, a second order finite volume approach on unstructured meshes, Comput. Methods Appl. Mech. Eng., 198 (2009), pp. 2520–2538.CrossRefGoogle Scholar
[9]Castro Díaz, M.J., Fernández-Nieto, E.D., De Luna, T. Morales, Narbona-Reina, G. and Parès, C., A HLLC scheme for nonconservative hyperbolic problems, Application to turbidity currents with sediment transport, ESAIM: M2AN, 47 (2013), pp. 1–32.CrossRefGoogle Scholar
[10]Castro Díaz, M., Lefloch, P.G., Muñoz-Ruiz, M.L. and Parès, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227(17) (2008), pp. 8107–8129.CrossRefGoogle Scholar
[11]Chalons, C., Transport-Equilibrium schemes for computing nonclassical shocks, scalar conservation laws, Numer. Methods Partial Differential Equations, 24(4) (2008), pp. 1127–1147.CrossRefGoogle Scholar
[12]Chalons, C. and Coquel, F., Euler equations with several independent pressure laws and entropy satisfying explicit projection schemes, Math. Models Methods Appl. Sci., 16 (2006), pp. 1469–1504.Google Scholar
[13]Chalons, C. and Coquel, F., Numerical capture of shock solutions of nonconservative hyperbolic systems via kinetic functions, Anal. Simulation Fluid Dyn. Adv. Math. Fluid Mech., Birkhäuser, Basel, (2007), pp. 45–68.Google Scholar
[14]Chalons, C., Coquel, F., Godlewski, E., Raviart, P.A. and Seguin, N., Godunov-type schemes for hyperbolic systems with parameter dependent source, The case ofEuler system with friction, Math. Models Methods Appl. Sci., 20 (2010), pp. 2109–2166.Google Scholar
[15]Colombeau, J.F. and Le Roux, A.Y., Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys., 29 (1988), pp. 315.Google Scholar
[16]Colombeau, J.F. and Le Roux, A.Y., Noussair, A. and Perrot, B., Microscopic profiles of shock waves and ambiguities in multiplications of distributions, SIAM J. Numer. Anal., 26 (1989), pp. 871–883.Google Scholar
[17]Cordier, S., Le Minh, H. and De Luna, T. Morales, Bedload transport in shallow water models: why splitting (may) fail, how hyperbolicity (can) help, Adv. Water Resources, 34 (2011), pp. 980–989.Google Scholar
[18]Dal Maso, G., Lefloch, P.G. and Murat, F., Definition and weakstability of nonconservative products, J. Math. Pures Appl., 74 (1995), pp. 483–548.Google Scholar
[19]Einstein, H.A., The bed load function for sediment transportation in open channel ows, Technical Bulletin No. 1026, Soil Conservation Service, US Department of Agriculture, (1950), pp. 1–71.Google Scholar
[20]Engelund, F. and Fredsoe, J., A sediment transport model for straight alluvial channels, Nordic Hydrol, 7 (1976), pp. 294–298.Google Scholar
[21]Exner, F.M., Über die Wechselwirkung zwishen Wasser und Geschiebe in Flüssen, Sitzungsber. Akad. Wiss. Wien, Math. Naturwiss. Kl., Abt. 2A, 134 (1925), pp. 165–1801.Google Scholar
[22]Gallice, G., Entropic Godunov-type schemes for hyperbolic systems with source term, C. R. Math. Acad. Sci. Paris, 334(8) (2002), pp. 713–716.Google Scholar
[23]Ghidaglia, J.M., Kumbaro, A. and Le Coq, G., On the numerical solution to two fluid models via a cell centered finite volume method, Eur. J. Mech. B Fluids, 20 (2001), pp. 841–867.Google Scholar
[24]Grass, A.J., Sediments transport by waves and currents, SERC London, Cent Mar Technol Report, No. FL29,1981.Google Scholar
[25]Godlewski, E. and Raviart, P.A., Numerical approximation of hyperbolic systems of conservation laws, Appl. Math. Sci., 118 (1996).Google Scholar
[26]Harten, A., Lax, P.D. and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), pp. 35–61.Google Scholar
[27]Hou, T.Y. and Lefloch, P.G., Why nonconservative schemes converge to wrong solutions, Error analysis, Math. Comput., 26 (1994), pp. 497–530.CrossRefGoogle Scholar
[28]Lefloch, P.G. and Tzavaras, A.E., Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal., 30(6) (1999), pp. 1309–1342.Google Scholar
[29]Leveque, R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, 2002.Google Scholar
[30]Meyer-Peter, E. and Muller, R., Formulas for bed-load transport, Proceedings of the 2nd Meeting of the International Association for Hydraulic Structures Research, (1948), pp. 39–64.Google Scholar
[31]Nielsen, P., Coastal bottom boundary layers and sediment transport, Advanced Series on Ocean Engeneering, 4 (1992).CrossRefGoogle Scholar
[32]Parès, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal., 44 (2006), pp. 300–321.Google Scholar
[33]Parès, C. and Castro, M.J., On the well-balance property of Roe’s method for nonconservative hyperbolic systems, Applications to shallow-water systems, Math. Model. Numer. Anal., 38 (2004), pp. 821–852.Google Scholar
[34]Raviart, P.A. and Sainsaulieu, L., A nonconservative hyperbolic system modelling spray dynamics, Part 1. Solution of the Riemann problem, Math. Models Methods App. Sci., 5(3) (1995), pp. 297–333.Google Scholar
[35]Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), pp. 237–263.Google Scholar
[36]Volpert, A.I., The space BV and quasilinear equations, Math. USSR Sbornik, 73(115) (1967), pp. 225–267.CrossRefGoogle Scholar