Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 33-36, 2009, Pages 2520-2538

  Castro Diaz M J ^a   Fernandez Nieto E D ^b   Ferreiro, A M ^c   Pares C ^a  

^a UNIVERSITY OF MÁLAGA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

^c UNIVERSITY OF A CORUÑA   (Spain)

Author keywords

Bedload sediment transport; Finite volume method; High order schemes; Nonconservative hyperbolic system; Shallow water; Unstructured meshes; Well balanced

Indexed keywords

BEDLOAD SEDIMENT TRANSPORT; HIGH ORDER SCHEMES; NONCONSERVATIVE HYPERBOLIC SYSTEM; SHALLOW WATER; UNSTRUCTURED MESHES;

EROSION; FINITE VOLUME METHOD; SEDIMENTOLOGY;

SEDIMENT TRANSPORT;

EID: 65849486600     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.03.001     Document Type: Article

References (39)

1. Anastasiou K., and Chan C.T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes. Numer. Methods Fluids 24 (1997) 1225-1245
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 J.M., and Parés C. 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 (2006) 1103-1134
4. Castro Di{dotless}́az 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 3 (2008) 299-316
5. M.J. Castro Di{dotless}́az, E.D. Fernández Nieto, A.M. Ferreiro, A. Garci{dotless}́a Rodri{dotless}́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., Maci{dotless}́a J., and 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 (2001) 107-127
7. Chien N. The present status of research on sediment transport. Trans. ASCE 121 (1956) 833-868
8. De Vrien H.J. 2DH mathematical modelling of morphological evolutions in shallow water. Coast. Engrg. 11 (1987) 1-27
9. Dumbser M., and Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221 2 (2007) 693-723
10. Dumbser M., Käser M., Titarev V.A., and Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226 (2007) 204-243
11. Dumbser M., Balsara D.S., Toro E.F., and 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 (2008) 8209-8253
12. Denys Dutykh, Modélisation mathématique des tsunamis, Ph.D. Thesis, École Normale Supérieure de Cachan, 2007.
13. Fowler A.C., Kopteva N., and Oakley C. The formation of river channels. SIAM J. Appl. Math. 67 4 (2007) 1016-1040
14. A.J. Grass, Sediments transport by waves and currents, SERC London Cent. Mar. Technol., Report No. FL29, 1981.
15. Harten A., and Hyman J.M. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50 (1983) 235-269
16. P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Inst. Math. Appl., Minneapolis, Preprint 593, 1989.
17. Hu C., and Shu C.-W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150 (1999) 97-127
18. J. Hudson, Numerical technics for morphodynamic modelling. Thesis Doctoral. University of Whiteknights, 2001.
19. Jiang G., and Shu C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1996) 202-228
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 (1972) 68-77
21. Liu X.D., Osher S., and Chan T. Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115 (1994) 200-212
22. Lyn D.A., and Altinakar M. St. Venant-Exner equations for near-critical and transcritical flows. J. Hydraul. Engrg. 128 6 (2002) 579-587
23. Marquina A. Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws. SIAM J. Sci. Comput. 15 (1994) 892-915
24. Dal Maso G., LeFloch P.G., and Murat F. Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483-548
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., and Natvig J. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499
28. Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300-321
29. Parés C., and 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 (2004) 821-852
30. Joachim Schroll H., and Svensson F. A bi-hyperbolic finite volume method on quadrilateral meshes. J. Sci. Comput. 26 2 (2006) 237-260
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., and Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77 (1998) 439-471
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., and Bokhovea O. A discontinuous Galerkin finite element model for river bed evolution under shallow flows. CMAME 197 33-40 (2008) 2930-2947
35. Toumi I. A weak formulation of Roe's approximate Riemann Solver. J. Comput. Phys. 102 2 (1992) 360-373
36. Van Leer B. Towards the ultimate conservative difference scheme v: a second order sequel to Godunov' method. J. Comput. Phys. 32 (1979) 101-136
37. Van Leer B. Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes. Commun. Comput. Phys. 1 (2006) 192-206
38. Van Rijn L.C. Sediment transport (I): bed load transport. J. Hydraul. Div., Proc. ASCE 110 (1984) 1431-1456
39. G. Walz, Romberg type cubature over arbitrary triangles, Mannheimer Mathem. Manuskripte Nr. 225, Mannhein, 1997.