Simple, accurate, and efficient revisions to MacCormack and Saulyev schemes: high Péclet numbers. (English) Zbl 1114.65103

Summary: Stream water quality modeling often involves numerical methods to solve the dynamic one-dimensional advection-dispersion-reaction equations (ADRE). There are numerous explicit and implicit finite difference schemes for solving these problems, and two commonly used schemes are the MacCormack and Saulyev schemes.
This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Péclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank-Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes are tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances are considered with respect to stability, accuracy, and computational efficiency.
Results indicate that both the modified Saulyev and the MacCormack schemes, which are named the Saulyev\(_{c}\) and MacCormack\(_{c}\) schemes, respectively, greatly improve the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyev\(_{c}\) scheme. The MacCormack\(_{c}\) scheme demonstrates increased time consumption but is still much faster than implicit schemes.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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[1] Ames, W.F., Numerical methods for partial differential equations, (1977), Academic Press New York, 365 pp · Zbl 0219.35007
[2] Chapra, S.C., Surface water-quality modeling, (1997), McGraw-Hill New York, 844 pp
[3] Chen, J.Y.; Ko, C.H.; Bhattacharjee, S.; Elimelech, M., Role of spatial distribution of porous medium surface charge heterogeneity in colloid transport, Colloids and surfaces A: physicochemical and engineering aspects, 191, 1-2, 3-15, (2001)
[4] Dehghan, M., Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition, Applied mathematics and computation, 147, 2, 321-331, (2004) · Zbl 1033.65068
[5] Dehghan, M., Weighted finite difference techniques for the one-dimensional advection – diffusion equation, Applied mathematics and computation, 147, 2, 307-319, (2004) · Zbl 1034.65069
[6] Dehghan, M., Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Applied numerical mathematics, 52, 1, 39-62, (2005) · Zbl 1063.65079
[7] Gerald, C.F.; Wheatley, P.O., Applied numerical analysis, (2004), Pearson/Addison-Wesley Boston, 609 pp
[8] Jury, W.A.; Roth, K., Transfer functions and solute movement through soil: theory and applications, (1990), Birkhäuser Verlag Basel, Boston, 226 pp
[9] Leonard, B.P., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Computer methods in applied mechanics and engineering, 19, 59-98, (1979) · Zbl 0423.76070
[10] Lin, Y.C.; Chang, M.S.; Medina, J.M.A., A methodology for solute transport in unsteady, nonuniform streamflow with subsurface interaction, Advances in water resources, 28, 8, 871-883, (2005)
[11] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations : an introduction, (1994), Cambridge University Press Cambridge, New York, 227 pp · Zbl 0811.65063
[12] O’Loughlin, E.M.; Bowmer, K.H., Dilution and decay of aquatic herbicides in flowing channels, Journal of hydrology, 26, 3-4, 217-235, (1975)
[13] Sousa, E.; Sobey, I., On the influence of numerical boundary conditions, Applied numerical mathematics, 41, 2, 325-344, (2002) · Zbl 0996.65082
[14] Stamou, A.I., Improving the numerical modeling of river water quality by using high order difference schemes, Water research, 26, 12, 1563-1570, (1992)
[15] Thomee, V., From finite differences to finite elements: a short history of numerical analysis of partial differential equations, Journal of computational and applied mathematics, 128, 1-2, 1-54, (2001) · Zbl 0977.65001
[16] W. Zeng, A model for understanding and managing the impacts of sediment behavior on river water quality, Ph.D. thesis, University of GA, Athens, GA, 2000, 244 pp. Available from: <http://getd.galib.uga.edu/public/zeng_wei_200012_phd/zeng_wei_200012_phd.pdf>.
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