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Journal of Applied Mathematics
Volume 2014, Article ID 274263, 8 pages
http://dx.doi.org/10.1155/2014/274263
Research Article

Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream

1Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand
2Centre of Excellence in Mathematics, Commission on Higher Education (CHE), Si Ayutthaya Road, Bangkok 10400, Thailand

Received 1 September 2013; Revised 17 December 2013; Accepted 17 December 2013; Published 23 February 2014

Academic Editor: Luís Godinho

Copyright © 2014 Nopparat Pochai. 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. N. Pochai, S. Tangmanee, L. J. Crane, and J. J. H. Miller, “A mathematical model of water pollution control using the finite element method,” Proceedings in Applied Mathematics and Mechanics, vol. 6, no. 1, pp. 755–756, 2006. View at Google Scholar
  2. J. Y. Chen, C. Ko, S. Bhattacharjee, and M. Elimelech, “Role of spatial distribution of porous medium surface charge heterogeneity in colloid transport,” Colloids and Surfaces A, vol. 191, no. 1-2, pp. 3–15, 2001. View at Publisher · View at Google Scholar · View at Scopus
  3. G. Li and C. R. Jackson, “Simple, accurate, and efficient revisions to MacCormack and Saulyev schemes: high Peclet numbers,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 610–622, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. M. O'Loughlin and K. H. Bowmer, “Dilution and decay of aquatic herbicides in flowing channels,” Journal of Hydrology, vol. 26, no. 3-4, pp. 217–235, 1975. View at Google Scholar · View at Scopus
  5. M. Dehghan, “Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 321–331, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. I. Stamou, “Improving the numerical modeling of river water quality by using high order difference schemes,” Water Research, vol. 26, no. 12, pp. 1563–1570, 1992. View at Publisher · View at Google Scholar · View at Scopus
  7. P. Tabuenca, J. Vila, J. Cardona, and A. Samartin, “Finite element simulation of dispersion in the Bay of Santander,” Advances in Engineering Software, vol. 28, no. 5, pp. 313–332, 1997. View at Google Scholar · View at Scopus
  8. N. Pochai, “A numerical computation of the non-dimensional form of a non-linear hydrodynamic model in a uniform reservoir,” Journal of Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 463–466, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. N. Pochai, S. Tangmanee, L. J. Crane, and J. J. H. Miller, “A water quality computation in the uniform channel,” Journal of Interdisciplinary Mathematics, vol. 11, no. 6, pp. 803–814, 2008. View at Google Scholar
  10. N. Pochai, “A numerical computation of a non-dimensional form of stream water quality model with hydrodynamic advection-dispersion-reaction equations,” Journal of Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 666–673, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, 2nd edition, 1977. View at MathSciNet
  12. S. C. Chapra, Surface Water-Quality Modeling, McGraw-Hill, 1997.
  13. N. Pochai, “A numerical treatment of nondimensional form of water quality model in a nonuniform flow stream using Saulyev scheme,” Mathematical Problems in Engineering, vol. 2011, Article ID 491317, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Karahan, “A third-order upwind scheme for the advection-diffusion equation using spreadsheets,” Advances in Engineering Software, vol. 38, no. 10, pp. 688–697, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. M.-S. Liou and C. J. Steffen Jr., “A new flux splitting scheme,” Applied Mathematics and Computation, vol. 107, no. 1, pp. 23–39, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Mazzia, L. Bergamaschi, C. N. Dawson, and M. Putti, “Godunov mixed methods on triangular grids for advection-dispersion equations,” Computational Geosciences, vol. 6, no. 2, pp. 123–139, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. Dawson, “Godunov-mixed methods for advection-diffusion equations in multidimensions,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1315–1332, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. K. Alhumaizi, “Flux-limiting solution techniques for simulation of reaction-diffusion-convection system,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 6, pp. 953–965, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. N. Happenhofer, O. Koch, F. Kupka, and F. Zaussinger, “Total variation diminishing implicit Runge-Kutta methods for dissipative advection-diffusion problems in astrophysics,” Proceedings in Applied Mathematics and Mechanics, vol. 11, pp. 777–778, 2011. View at Google Scholar
  20. H. Ninomiya and K. Onishi, Flow Analysis Using a PC, CRC Press, 1991.
  21. A. R. Mitchell, Computational Methods in Partial Differential Equations, John Wiley & Sons, 1969. View at MathSciNet
  22. B. Bradie, A Friendly Introduction to Numerical Analysis, Prentice Hall, 2005.