Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T20:41:20.409Z Has data issue: false hasContentIssue false
  • English
  • Français

A HLLC scheme for nonconservative hyperbolic problems.Application to turbidity currents with sediment transport

Published online by Cambridge University Press:  31 July 2012

Manuel Jesús Castro Díaz ,
Enrique Domingo Fernández-Nieto ,
Tomás Morales de Luna ,
Gladys Narbona-Reina  and
Carlos Parés
Manuel Jesús Castro Díaz
Affiliation:
Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málga, Campus de Teatinos, s/n, 29071 Málaga, Spain. [email protected]; [email protected]
Enrique Domingo Fernández-Nieto
Affiliation:
Dpto. Matemática Aplicada I, ETS Arquitectura, Universidad de Sevilla, Avda. Reina Mercedes No. 2, 41012 Sevilla, Spain; [email protected]; [email protected]
Tomás Morales de Luna
Affiliation:
Dpto. de Matemáticas, Universidad de Córdoba, Campus de Rabanales, 14071 Córdoba, Spain; [email protected]
Gladys Narbona-Reina
Affiliation:
Dpto. de Matemáticas, Universidad de Córdoba, Campus de Rabanales, 14071 Córdoba, Spain; [email protected]
Carlos Parés
Affiliation:
Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málga, Campus de Teatinos, s/n, 29071 Málaga, Spain. [email protected]; [email protected]
Get access

Abstract

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-typeapproximate Riemann solver for a hyperbolic nonconservative PDE system arising in aturbidity current model. The main difficulties come from the nonconservative nature of thesystem. A general strategy to derive simple approximate Riemann solvers fornonconservative systems is introduced, which is applied to the turbidity current model toobtain two different HLLC solvers. Some results concerning the non-negativity preservingproperty of the corresponding numerical methods are presented. The numerical resultsprovided by the two HLLC solvers are compared between them and also with those obtainedwith a Roe-type method in a number of 1d and 2d test problems. This comparison shows that,while the quality of the numerical solutions is comparable, the computational cost of theHLLC solvers is lower, as only some partial information of the eigenstructure of thematrix system is needed.

Keywords

Well-balancedfinite volume methodpath-conservativesimple Riemann solverHLLC
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 1 - 32
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abgrall, R. and Karni, S., A comment on the computation of non-conservative products. J. Comput. Phys. 229 (2010) 2759–2763. Google Scholar
Altinaker, M.S., Graf, W.H. and Hopfinger, E., Flow structure in turbidity currents. J. Hydr. Res. 34 (1996) 713–718. Google Scholar
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, in Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).
Bradford, S.F. and Katopodes, N.D., Hydrodynamics of turbid underflows. i : Formulation and numerical analysis. J. Hydr. Eng. 125 (1999) 1006–1015. Google Scholar
Castro, M.J., LeFloch, P.G., Muñoz-Ruiz, M.L. and Parés, C., Why many theories of shock waves are necessary : Convergence error in formally path-consistent schemes. J. Comput. Phys. 227 (2008) 8107–8129. Google Scholar
Castro, M.J., Fernández-Nieto, E.D., Ferreiro, A.M., García-Rodríguez, J.A. and Parés, C., High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems. J. Sci. Comput. 39 (2009) 67–114. Google Scholar
Castro Díaz, M., Fernéndez-Nieto, E. and Ferreiro, A., Sediment transport models in shallow water equations and numerical approach by high order finite volume methods. Comput. Fluids 37 (2008) 299–316. Google Scholar
Castro Díaz, M.J., Fernández-Nieto, E.D., Ferreiro, A.M. and Parés, 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) 2520–2538. Google Scholar
Cordier, S., Le, M. and Morales de Luna, T., Bedload transport in shallow water models : Why splitting (may) fail, how hyperbolicity (can) help. Adv. Water Resour. 34 (2011) 980–989. Google Scholar
Dal Maso, G., Lefloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. Google Scholar
F. Exner, Über die wechselwirkung zwischen wasser und geschiebe in flüssen. Sitzungsber., Akad. Wissenschaften IIa (1925).
Fernández-Nieto, E.D., Modelling and numerical simulation of submarine sediment shallow flows : transport and avalanches. Bol. Soc. Esp. Mat. Apl. S􏿻 MA 49 (2009) 83–103. Google Scholar
Fowler, A.C., Kopteva, N. and Oakley, C., The formation of river channels. SIAM J. Appl. Math. 67 (2007) 1016–1040. Google Scholar
Gallardo, J., Ortega, S., de la Asunción, M. and Mantas, J., Two-Dimensional compact third-order polynomial reconstructions. solving nonconservative hyperbolic systems using GPUs. J. Sci. Comput. 48 (2011) 141–163. Google Scholar
A. Grass, Sediment transport by waves and currents. SERC London Cent. Mar. Technol. Report No. FL29 (1981).
A. Harten, P.D. Lax and B. van Leer, On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983).
Hou, T.Y. and Le Floch, P.G., Why nonconservative schemes converge to wrong solutions : error analysis. Math. Comput. 62 (1994) 497–530. Google Scholar
Khan, S.M., Imran, J., Bradford, S. and Syvitski, J., Numerical modeling of hyperpycnal plume. Mar. Geol. 222-223 (2005) 193–211. Google Scholar
Kubo, Y., Experimental and numerical study of topographic effects on deposition from two-dimensional, particle-driven density currents. Sediment. Geol. 164 (2004) 311–326. Google Scholar
Kubo, Y. and Nakajima, T., Laboratory experiments and numerical simulation of sediment-wave formation by turbidity currents. Mar. Geol. 192 (2002) 105–121. Google Scholar
Lyn, D.A. and Altinakar, M., St. Venant-Exner equations for Near-Critical and transcritical flows. J. Hydr. Eng. 128 (2002) 579–587. Google Scholar
E. Meyer-Peter, and R. Müller, Formulas for bed-load transport, in 2 nd meeting IAHSR. Stockholm, Sweden (1948) 1–26.
Morales de Luna, T., Castro Díaz, M.J., Parés Madroñal, C. and Fernández Nieto, E.D., On a shallow water model for the simulation of turbidity currents. Commun. Comput. Phys. 6 (2009) 848–882. Google Scholar
Morales de Luna, T., Castro Díaz, M.J. and Parés Madroñal, C., A duality method for sediment transport based on a modified Meyer-Peter & Müller model. J. Sci. Comput. 48 (2010) 258–273. Google Scholar
Morris, P.H. and Williams, D.J., Relative celerities of mobile bed flows with finite solids concentrations. J. Hydr. Eng. 122 (1996) 311–315. Google Scholar
Muñoz Ruiz, M.L. and Parés, C., On the convergence and Well-Balanced property of Path-conservative numerical schemes for systems of balance laws. J. Sci. Comput. 48 (2011) 274–295. Google Scholar
P. Nielsen, Coastal Bottom Boundary Layers and Sediment Transport. World Scientific Pub. Co. Inc. (1992).
Parés, C., Numerical methods for nonconservative hyperbolic systems : a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300–321 (electronic). Google Scholar
Parés, C. and Muñoz Ruiz, M.L., On some difficulties of the numerical approximation of nonconservative hyperbolic systems. Bol. Soc. Esp. Mat. Apl. 47 (2009) 23–52. Google Scholar
Parker, G., Fukushima, Y. and Pantin, H.M., Self-accelerating turbidity currents. J. Fluid Mech. 171 (1986) 145–181. Google Scholar
Toro, E.F., Spruce, M. and Speares, W., Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4 (1994) 25–34. Google Scholar
E.F. Toro, Shock-capturing methods for free-surface shallow flows. John Wiley (2001).
Van Rijn, L., Sediment transport : bed load transport. J. Hydr. Eng. 110 (1984) 1431–1456.Google Scholar