Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014, Article ID 257915, 15 pages
http://dx.doi.org/10.1155/2014/257915
Research Article

Development of a Cell-Centered Godunov-Type Finite Volume Model for Shallow Water Flow Based on Unstructured Mesh

1Institute of Hydraulic Structures and Water Environment, Zhejiang University, Hangzhou 310058, China
2Institute of Physical Oceanography, Ocean College, Zhejiang University, Hangzhou 310058, China
3State Key Laboratory of Satellite Ocean Environment Dynamics, The Second Institute of Oceanography, Hangzhou 310012, China

Received 6 December 2013; Accepted 15 April 2014; Published 28 May 2014

Academic Editor: Yonghong Wu

Copyright © 2014 Gangfeng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. J. Fennema and M. H. Chaudhry, “Explicit methods for 2-D transient free-surface flows,” Journal of Hydraulic Engineering, vol. 116, no. 8, pp. 1013–1034, 1990. View at Google Scholar · View at Scopus
  2. T. Molls and M. H. Chaudhry, “Depth-averaged open-channel flow model,” Journal of Hydraulic Engineering, vol. 121, no. 6, pp. 453–465, 1995. View at Google Scholar · View at Scopus
  3. F. Alcrudo and P. Garcia-Navarro, “High-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations,” International Journal for Numerical Methods in Fluids, vol. 16, no. 6, pp. 489–505, 1993. View at Google Scholar · View at Scopus
  4. K. Anastasiou and C. T. Chan, “Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes,” International Journal for Numerical Methods in Fluids, vol. 24, pp. 1225–1245, 1997. View at Google Scholar
  5. L. Begnudelli and B. F. Sanders, “Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying,” Journal of Hydraulic Engineering, vol. 132, no. 4, pp. 371–384, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. D. H. Zhao, H. W. Shen, G. Q. Tabios III, J. S. Lai, and W. Y. Tan, “Finite-volume two-dimensional unsteady-flow model for river basins,” Journal of Hydraulic Engineering, vol. 120, no. 7, pp. 863–883, 1994. View at Google Scholar · View at Scopus
  7. H. Liu, J. G. Zhou, and R. Burrows, “Lattice Boltzmann simulations of the transient shallow water flows,” Advances in Water Resources, vol. 33, no. 4, pp. 387–396, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Chippada, C. N. Dawson, M. L. Martinez, and M. F. Wheeler, “A Godunov-type finite volume method for the system of shallow water equations,” Computer Methods in Applied Mechanics and Engineering, vol. 151, no. 1-2, pp. 105–129, 1998. View at Google Scholar · View at Scopus
  9. Q. Liang and F. Marche, “Numerical resolution of well-balanced shallow water equations with complex source terms,” Advances in Water Resources, vol. 32, no. 6, pp. 873–884, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. Q. Liang, “Flood simulation using a well-balanced shallow flow model,” Journal of Hydraulic Engineering, vol. 136, no. 9, pp. 669–675, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Begnudelli and B. F. Sanders, “Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying,” Journal of Hydraulic Engineering, vol. 132, no. 4, pp. 371–384, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. S. F. Bradford and B. F. Sanders, “Finite-volume model for shallow-water flooding of arbitrary topography,” Journal of Hydraulic Engineering, vol. 128, no. 3, pp. 289–298, 2002. View at Publisher · View at Google Scholar · View at Scopus
  13. L. Song, J. Zhou, J. Guo, Q. Zou, and Y. Liu, “A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain,” Advances in Water Resources, vol. 34, no. 7, pp. 915–932, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. A. I. Delis, M. Kazolea, and N. A. Kampanis, “A robust high-resolution finite volume scheme for the simulation of long waves over complex domains,” International Journal for Numerical Methods in Fluids, vol. 56, no. 4, pp. 419–452, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. G. Zhou, D. M. Causon, C. G. Mingham, and D. M. Ingram, “The surface gradient method for the treatment of source terms in the shallow-water equations,” Journal of Computational Physics, vol. 168, no. 1, pp. 1–25, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Hou, F. Simons, and R. Hinkelmann, “A 2D well-balanced shallow flow model for unstructured grids with novel slope source treatment,” Advances in Water Resources, vol. 52, pp. 107–131, 2013. View at Publisher · View at Google Scholar
  17. J. M. Greenberg and A. Y. Leroux, “A well-balanced scheme for the numerical processing of source terms in hyperbolic equations,” SIAM Journal on Numerical Analysis, vol. 33, no. 1, pp. 1–16, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. Bermúdez and M. E. Vazquez, “Upwind methods for hyperbolic conservation laws with source terms,” Computers and Fluids, vol. 23, no. 8, pp. 1049–1071, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. G. Zhou, D. M. Causon, D. M. Ingram, and C. G. Mingham, “Numerical solutions of the shallow water equations with discontinuous bed topography,” International Journal for Numerical Methods in Fluids, vol. 38, no. 8, pp. 769–788, 2002. View at Publisher · View at Google Scholar · View at Scopus
  20. Q. Liang and A. G. L. Borthwick, “Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography,” Computers and Fluids, vol. 38, no. 2, pp. 221–234, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. E. Audusse, F. Bouchut, M. Bristeau, R. Klein, and B. Perthame, “A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows,” SIAM Journal on Scientific Computing, vol. 25, no. 6, pp. 2050–2065, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. F. Marche, P. Bonneton, P. Fabrie, and N. Seguin, “Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes,” International Journal for Numerical Methods in Fluids, vol. 53, no. 5, pp. 867–894, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. L. Xing, X. X. Zhang, and C. W. Shu, “Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations,” Advances in Water Resources, vol. 33, no. 12, pp. 1476–1493, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Kurganov and G. Petrova, “A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system,” Communications in Mathematical Sciences, vol. 5, no. 1, pp. 133–160, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Kurganov and D. Levy, “Central-upwind schemes for the Saint-Venant system,” Mathematical Modelling and Numerical Analysis, vol. 36, no. 3, pp. 397–425, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Bryson, Y. Epshteyn, A. Kurganov, and G. Petrova, “Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 45, no. 3, pp. 423–446, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Singh, M. S. Altinakar, and Y. Ding, “Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme,” Advances in Water Resources, vol. 34, no. 10, pp. 1366–1375, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. M. E. Hubbard, “Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids,” Journal of Computational Physics, vol. 155, no. 1, pp. 54–74, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. R. Shewchuk, “Triangle: engineering a 2D quality mesh generator and Delaunay triangulator,” in Applied Computational Geometry: Towards Geometric Engineering, vol. 1148 of Lecture Notes in Computer Science, pp. 203–222, 1996. View at Google Scholar
  30. A. Duran, Q. Liang, and F. Marche, “On the well-balanced numerical discretization of shallow water equations on unstructured meshes,” Journal of Computational Physics, vol. 235, pp. 565–586, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. N. Goutal and F. Maurel, Eds., “Proceedings of the 2nd workshop on dam-break wave simulation,” Tech. Rep. HE 43/97/016/B, Départment Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale Electricité de France, Paris, France, 1997. View at Google Scholar
  32. W. C. Thacker, “Some exact solutions to the nonlinear shallow-water wave equations,” Journal of Fluid Mechanics, vol. 107, pp. 499–508, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. Guan, N. G. Wright, and P. A. Sleigh, “A robust 2D shallow water model for solving flow over complex topography using homogenous flux method,” International Journal for Numerical Methods in Fluids, vol. 73, no. 3, pp. 225–249, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  34. C. V. Bellos, J. V. Soulis, and J. G. Sakkas, “Experimental investigation of two-dimensional dam-break induced flows,” Journal of Hydraulic Research, vol. 30, no. 1, pp. 47–75, 1992. View at Google Scholar · View at Scopus
  35. J. Q. Xia, B. L. Lin, R. A. Falconer, and G. Q. Wang, “Modelling dam-break flows over mobile beds using a 2D coupled approach,” Advances in Water Resources, vol. 33, no. 2, pp. 171–183, 2010. View at Publisher · View at Google Scholar · View at Scopus
  36. S. N. Kuiry, D. Sen, and Y. Ding, “A high-resolution shallow water model using unstructured quadrilateral grids,” Computers & Fluids, vol. 68, pp. 16–28, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  37. IMPACT, EC Contract EVG1-CT-2001-00037. Investigation of Extreme Flood Processes and Uncertainties, 2004, http://www.impact-project.net/.
  38. S. S. Frazão, “Experiments of dam-break wave over a triangular bottom sill,” Journal of Hydraulic Research, vol. 45, pp. 19–26, 2007. View at Google Scholar · View at Scopus
  39. J. M. Hou, Q. H. Liang, F. Simons, and R. Hinkelmann, “A stable 2D unstructured shallow flow model for simulations of wetting and drying over rough terrains,” Computers & Fluids, vol. 82, pp. 132–147, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  40. Y. Wang, Q. Liang, G. Kesserwani, and J. W. Hall, “A 2D shallow flow model for practical dam-break simulations,” Journal of Hydraulic Research, vol. 49, no. 3, pp. 307–316, 2011. View at Publisher · View at Google Scholar · View at Scopus
  41. T. H. Yoon and S. Kang, “Finite volume model for two-dimensional shallow water flows on unstructured grids,” Journal of Hydraulic Engineering, vol. 130, no. 7, pp. 678–688, 2004. View at Publisher · View at Google Scholar · View at Scopus
  42. J. M. Hervouet and A. Petitjean, “Malpasset dam-break revisited with two-dimensional computations,” Journal of Hydraulic Research, vol. 37, no. 6, pp. 777–788, 1999. View at Google Scholar · View at Scopus
  43. A. Valiani, V. Caleffi, and A. Zanni, “Case study: malpasset dam-break simulation using a two-dimensional finite volume method,” Journal of Hydraulic Engineering, vol. 128, no. 5, pp. 460–472, 2002. View at Publisher · View at Google Scholar · View at Scopus
  44. M. L. Zhang and W. M. Wu, “A two dimensional hydrodynamic and sediment transport model for dam break based on finite volume method with quadtree grid,” Applied Ocean Research, vol. 33, no. 4, pp. 297–308, 2011. View at Publisher · View at Google Scholar · View at Scopus