Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2013), Article ID 274263, 8 pages.

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

Nopparat Pochai

Full-text: Open access

Abstract

Two mathematical models are used to simulate water quality in a nonuniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the elevation of water. The second model is the dispersion model that provides the pollutant concentration field. Both models are formulated in one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. A modified MacCormack method is subsequently employed in the second model. This paper proposes a simply remarkable alteration to the MacCormack method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed modified MacCormack scheme does improve the prediction accuracy compared to that of the traditional MacCormack method.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 274263, 8 pages.

Dates
First available in Project Euclid: 1 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1412183190

Digital Object Identifier
doi:10.1155/2014/274263

Mathematical Reviews number (MathSciNet)
MR3176814

Zentralblatt MATH identifier
07010584

Citation

Pochai, Nopparat. Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream. J. Appl. Math. 2014, Special Issue (2013), Article ID 274263, 8 pages. doi:10.1155/2014/274263. https://projecteuclid.org/euclid.jam/1412183190


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References

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, 2nd edition, 1977.
  • S. C. Chapra, Surface Water-Quality Modeling, McGraw-Hill, 1997.
  • 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.
  • 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.
  • 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.
  • 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.
  • C. Dawson, “Godunov-mixed methods for advection-diffusion equations in multidimensions,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1315–1332, 1993.
  • 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.
  • 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.
  • H. Ninomiya and K. Onishi, Flow Analysis Using a PC, CRC Press, 1991.
  • A. R. Mitchell, Computational Methods in Partial Differential Equations, John Wiley & Sons, 1969.
  • B. Bradie, A Friendly Introduction to Numerical Analysis, Prentice Hall, 2005. \endinput