zbMATH — the first resource for mathematics

Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes. (English) Zbl 1228.76091
Summary: In this paper, we study the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modeled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a nonconservative hyperbolic system. To discretize it, first we consider a Roe-type first order scheme as well as a variant based on the use of flux limiters. These first order schemes are then extended to second order accuracy by means of a new MUSCL-type reconstruction operator on unstructured meshes. Finally, some numerical tests are presented.

76M12 Finite volume methods applied to problems in fluid mechanics
76T20 Suspensions
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76A05 Non-Newtonian fluids
PDF BibTeX Cite
Full Text: DOI
[1] Anastasiou, K.; Chan, C.T., Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes, Numer. methods fluids, 24, 1225-1245, (1997) · Zbl 0886.76064
[2] T. Barth, D. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA Paper 89-0366, 1989.
[3] Castro, M.J.; Gallardo, José M.; Parés, Carlos, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math. comput., 75, 255, 1103-1134, (2006) · Zbl 1096.65082
[4] Castro Dı´az, M.J.; Fernández Nieto, E.D.; Ferreiro, A.M., Sediment transport models in shallow water equations and numerical approach by high order finite volume methods, Comput. fluids, 37, 3, 299-316, (2008) · Zbl 1237.76082
[5] M.J. Castro Dı´az, E.D. Fernández Nieto, A.M. Ferreiro, A. Garcı´a Rodrı´guez, C. Parés, High order extension of Roe schemes for two dimensional nonconservative hyperbolic systems, J. Sci. Comput. doi:10.1007/s10915-008-9250-4.
[6] Castro, M.J.; Macı´a, J.; Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system, ESAIM-math. model. num., 35, 1, 107-127, (2001) · Zbl 1094.76046
[7] Chien, N., The present status of research on sediment transport, Trans. ASCE, 121, 833-868, (1956)
[8] De Vrien, H.J., 2DH mathematical modelling of morphological evolutions in shallow water, Coast. engrg., 11, 1-27, (1987)
[9] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. comput. phys., 221, 2, 693-723, (2007) · Zbl 1110.65077
[10] Dumbser, M.; Käser, M.; Titarev, V.A.; Toro, E.F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. comput. phys., 226, 204-243, (2007) · Zbl 1124.65074
[11] Dumbser, M.; Balsara, D.S.; Toro, E.F.; Munz, C.D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes, J. comput. phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[12] Denys Dutykh, Modélisation mathématique des tsunamis, Ph.D. Thesis, École Normale Supérieure de Cachan, 2007.
[13] Fowler, A.C.; Kopteva, N.; Oakley, C., The formation of river channels, SIAM J. appl. math., 67, 4, 1016-1040, (2007) · Zbl 1117.86002
[14] A.J. Grass, Sediments transport by waves and currents, SERC London Cent. Mar. Technol., Report No. FL29, 1981.
[15] Harten, A.; Hyman, J.M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. comput. phys., 50, 235-269, (1983) · Zbl 0565.65049
[16] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Inst. Math. Appl., Minneapolis, Preprint 593, 1989.
[17] Hu, Changqing; Shu, Chi-Wang, Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090
[18] J. Hudson, Numerical technics for morphodynamic modelling. Thesis Doctoral. University of Whiteknights, 2001.
[19] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[20] Kolgan, N.E., Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics, Uchenye zapiski tsagi [sci. notes central inst. aerodyn], 3, 6, 68-77, (1972)
[21] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially nonoscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076
[22] Lyn, D.A.; Altinakar, M., St. venant – exner equations for near-critical and transcritical flows, J. hydraul. engrg., 128, 6, 579-587, (2002)
[23] Marquina, A., Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws, SIAM J. sci. comput., 15, 892-915, (1994) · Zbl 0805.65088
[24] Dal Maso, G.; LeFloch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068
[25] E. Meyer-Peter, R. Müller, Formulas for bed-load transport, in: Rep. 2nd Meet. Int. Assoc. Hydraul. Struct. Res., Stockholm, 1948, pp. 39-64.
[26] Nielsen, P., Coastal bottom boundary layers and sediment transport, Advanced series on Ocean engineering, vol. 4, (1992), World Scientific Publishing Singapore
[27] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. comput. phys., 213, 474-499, (2006) · Zbl 1088.76037
[28] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. numer. anal., 44, 300-321, (2006) · Zbl 1130.65089
[29] Parés, C.; Castro, M.J., On the well-balance property of roe’s method for non conservative hyperbolic systems. applications to shallow-water systems, Esaim: m2an, 38, 5, 821-852, (2004) · Zbl 1130.76325
[30] Joachim Schroll, H.; Svensson, Fredrik, A bi-hyperbolic finite volume method on quadrilateral meshes, J. sci. comput., 26, 2, 237-260, (2006) · Zbl 1203.76096
[31] C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Report: 97-65, 1997.
[32] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 77, 439-471, (1998) · Zbl 0653.65072
[33] Richard Soulsby, Dynamics of marine sands. A manual for practical applications, Published by Thomas Telford Publications, Thomas Telford Services Ltd, 1997.
[34] Tassi, P.A.; Rhebergena, S.; Vionnetb, C.A.; Bokhovea, O., A discontinuous Galerkin finite element model for river bed evolution under shallow flows, Cmame, 197, 33-40, 2930-2947, (2008) · Zbl 1194.76143
[35] Toumi, I., A weak formulation of roe’s approximate Riemann solver, J. comput. phys., 102, 2, 360-373, (1992) · Zbl 0783.65068
[36] Van Leer, B., Towards the ultimate conservative difference scheme v: a second order sequel to godunov’ method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
[37] Van Leer, B., Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes, Commun. comput. phys., 1, 192-206, (2006) · Zbl 1114.76049
[38] Van Rijn, L.C., Sediment transport (I): bed load transport, J. hydraul. div., proc. ASCE, 110, 1431-1456, (1984)
[39] G. Walz, Romberg type cubature over arbitrary triangles, Mannheimer Mathem. Manuskripte Nr. 225, Mannhein, 1997.
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.