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Mathematical Development and Verification of a Finite Volume Model for Morphodynamic Flow Applications

Published online by Cambridge University Press:  03 June 2015

Fayssal Benkhaldoun ,
Mohammed Seaïd  and
Slah Sahmim
Fayssal Benkhaldoun*
Affiliation:
LAGA, Université Paris 13, 99 Av J.B. Clement, 93430 Villetaneuse, France
Mohammed Seaïd*
Affiliation:
School of Engineering and Computing Sciences, University of Durham, South Road, Durham DH1 3LE, UK
Slah Sahmim*
Affiliation:
Laboratoire D’ingénierie Mathématique, Ecole Polytechnique de Tunisie, B.P. 743-2078 La Marsa, Tunisia
*
Corresponding author. URL: http://www.math.univ-paris13.fr/∼fayssal/ Email: fayssal@math.univ-paris13.fr
Email: m.seaid@durham.ac.uk
Email: s.sahmim@gmail.com
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Abstract

The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension. The governing equations consist of two components, namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation. Based on different formulations of the morphodynamic equations, we propose a family of three finite volume methods. The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport. The obtained results for several morphodynamic problems are considered to be representative, and might be helpful for a fair rating of finite volume solution schemes, particularly in long time computations.

Keywords

65M0876B1576M12Morphodynamic modelshallow water equationssediment transportfinite volume methodwell-balanced discretization
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 4 , August 2011 , pp. 470 - 492
Copyright
Copyright © Global-Science Press 2011

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References

[1] Benkhaldoun, F., Elmahi, I. and Seaïd, M., Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes, Comput. Phys., 226 (2007), pp. 180–203.CrossRefGoogle Scholar
[2] Benkhaldoun, F., Sahmim, S. and Seaïd, M., A two-dimensional finite volume morpho-dynamic model on unstructured triangular grids, Int. J. Numer. Meth. Fluids., 63 (2001), pp. 1296–1327.CrossRefGoogle Scholar
[3] Benkhaldoun, F., Sahmim, S. and Seaïd, M., Solution of the sediment transport equations using a finite volume method based on sign matrix, SIAM J. Sci. Comput., 31 (2009), pp. 2866–2889.Google Scholar
[4] Bermúdez, A. and Vázquez, E. M., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids., 23 (1994), pp. 1049–1071.CrossRefGoogle Scholar
[5] Caleffi, V., Valiani, A. and Bernini, A., High-order balanced CWENO scheme for movable bed shallow water equations, Adv. Water. Res., 30 (2007), pp. 730–741.Google Scholar
[6] Crnjaric, N. Z., Vukovic, S. and Sopta, L., Extension of ENO and WENO schemes to one-dimensional sediment transport equations, Comput. Fluids., 33 (2004), pp. 31–56.Google Scholar
[7] Crotogino, A. and Holz, K. P., Numerical movable-bed models for practical engineering, Appl. Math. Model., 8 (1984), pp. 45–49.Google Scholar
[8] Delis, A. I. and Papoglou, I., Relaxation approximation to bed-load sediment transport, J. Comput. Appl. Math., 213 (2008), pp. 521–546.Google Scholar
[9] Hudson, J. and Sweby, P. K., Formations for numerically approximating hyperbolic systems governing sediment transport, J. Sci. Comput., 19 (2003), pp. 225–252.Google Scholar
[10] Grass, A. J., Sediment transport by waves and currents, SERC London Cent. Mar. Technol., Report No: FL29, 1981.Google Scholar
[11] Randall, J. L., Numerical Methods for Conservation Laws, Lect. Math. ETH Zürich., 1992.Google Scholar
[12] Liang, C., Kannan, R. and Wang, Z. J., A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids, Comput. Fluids., 38 (2009), pp. 254–265.CrossRefGoogle Scholar
[13] Long, W., Kirby, J. T. and Shao, Z., A numerical scheme for morphological bed level calculations, Coastal. Eng., 55 (2008), pp. 167–180.Google Scholar
[14] Masella, J. M., Faille, I. and Gallouët, T., On an approximate Godunov scheme, Int. J. Comput. Fluids. Dyn., 13 (1999), pp. 133–149.Google Scholar
[15] Pritchard, D. and Hogg, A. J., On sediment transport under dam-break flow, J. Fluid. Mech., 473 (2002), pp. 265–274.CrossRefGoogle Scholar
[16] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.Google Scholar
[17] Rosatti, G. and Fraccarollo, L., A well-balanced approach for flows over mobile-bed with high sediment-transport, J. Comput. Phys., 220 (2006), pp. 312–338.Google Scholar
[18] Sahmim, S., Benkhaldoun, F. and Alcrudo, F., A sign matrix based scheme for quasi-hyperbolic non-homogeneous PDEs with an analysis of the convergence stagnation problem, J. Comput. Phys., 226 (2007), pp. 1753–1783.CrossRefGoogle Scholar
[19] Sahmim, S., Un Schéma aux Volumes Finis avec Matrice Signe pour les Systémes Non Homogénes, Dissertation, University of Paris 13, 2005.Google Scholar
[20] Vázquez, M. E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys., 148 (1999), pp. 497–526.Google Scholar