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Mathematical Problems in Engineering
Volume 2015, Article ID 379281, 10 pages
http://dx.doi.org/10.1155/2015/379281
Research Article

Convergence Improved Lax-Friedrichs Scheme Based Numerical Schemes and Their Applications in Solving the One-Layer and Two-Layer Shallow-Water Equations

1State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
2Hydrology Bureau, Yangtze River Water Resource Commission, Wuhan 430010, China
3Yangtze River Scientific Research Institute, Wuhan 430015, China

Received 13 August 2015; Accepted 22 October 2015

Academic Editor: Maurizio Brocchini

Copyright © 2015 Xinhua Lu 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. P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Communications on Pure and Applied Mathematics, vol. 7, no. 1, pp. 159–193, 1954. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. A. Barmin, A. G. Kulikovskiy, and N. V. Pogorelov, “Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics,” Journal of Computational Physics, vol. 126, no. 1, pp. 77–90, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. G. Tóth and D. Odstrčil, “Comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems,” Journal of Computational Physics, vol. 128, no. 1, pp. 82–100, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, Springer, Berlin, Germany, 3rd edition, 2009.
  5. H. Nessyahu and E. Tadmor, “Non-oscillatory central differencing for hyperbolic conservation laws,” Journal of Computational Physics, vol. 87, no. 2, pp. 408–463, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. E. F. Toro and S. J. Billett, “Centred TVD schemes for hyperbolic conservation laws,” IMA Journal of Numerical Analysis, vol. 20, no. 1, pp. 47–79, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. P. D. Lax and B. Wendroff, “Systems of conservation laws,” Communications on Pure and Applied Mathematics, vol. 13, pp. 217–237, 1960. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, 2001.
  9. E. F. Toro, A. Hidalgo, and M. Dumbser, “FORCE schemes on unstructured meshes. I. Conservative hyperbolic systems,” Journal of Computational Physics, vol. 228, no. 9, pp. 3368–3389, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. Sussman, P. Smereka, and S. Osher, “A level set approach for computing solutions to incompressible two-phase flow,” Journal of Computational Physics, vol. 114, no. 1, pp. 146–159, 1994. View at Publisher · View at Google Scholar · View at Scopus
  11. C.-W. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes, II,” Journal of Computational Physics, vol. 83, no. 1, pp. 32–78, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. E. Audusse, F. Bouchut, M.-O. 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 · View at Scopus
  13. B. van Leer, “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method,” Journal of Computational Physics, vol. 32, no. 1, pp. 101–136, 1979. View at Publisher · View at Google Scholar · View at Scopus
  14. P. L. Roe and M. J. Baines, “Algorithms for advection and shock problems,” in 4th GAMM Conference on Numerical Methods in Fluid Mechanics, pp. 281–290, Vieweg, Braunschweig, Germany, 1982. View at Google Scholar
  15. A. Bermudez and M. E. Vazquez, “Upwind methods for hyperbolic conservation laws with source terms,” Computers & Fluids, vol. 23, no. 8, pp. 1049–1071, 1994. View at Publisher · View at Google Scholar · View at Scopus
  16. J. J. Stoker, Water Waves: The Mathematical Theory with Applications, Interscience Publishers, New York, NY, USA, 1957. View at MathSciNet
  17. X. Lu, B. Dong, B. Mao, and X. Zhang, “A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography,” Comptes Rendus Mécanique, vol. 343, no. 7-8, pp. 429–442, 2015. View at Publisher · View at Google Scholar
  18. B. Spinewine, V. Guinot, S. Soares-Frazão, and Y. Zech, “Solution properties and approximate Riemann solvers for two-layer shallow flow models,” Computers & Fluids, vol. 44, no. 1, pp. 202–220, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. F. Bouchut and T. M. de Luna, “An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 42, no. 4, pp. 683–698, 2008. View at Google Scholar
  20. M. Dudzinski and M. Lukáčová-Medvid’ová, “Well-balanced bicharacteristic-based scheme for multilayer shallow water flows including wet/dry fronts,” Journal of Computational Physics, vol. 235, pp. 82–113, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. F. Bouchut and V. Zeitlin, “A robust well-balanced scheme for multi-layer shallow water equations,” Discrete and Continuous Dynamical Systems—Series B, vol. 13, no. 4, pp. 739–758, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus