Abstract
Upwind-biased discrete approximations have a distinguished history in numerical fluid mechanics, dating at least to von Neumann and Richtmyer (1950). Lately, however, upwinding has come under fire in water resources engineering. Among the most effective critics of upwind techniques are Gresho and Lee (1980), who take umbrage at the smearing of steep gradients in solutions of partial differential equations. While this viewpoint has cogency, a blanket condemnation of upwinding would be injudicious. There exist fluid flows for which upstream-biased discretizations are not only valid but in fact mathematically more appropriate than central approximations having higher-order accuracy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allen, M.B. (to appear), ‘How upstream collocation works,“ Int. J. Num. Meth. Engrg.
Allen, M.B., and Pinder, C.F. (1983), “Collocation simulation of multiphase porous-medium flow,’ Soc. Pet. Eng. J. ( February ), 135–142.
Aziz, K., and Settari, A. (1979), Petroleum Reservoir Simulation, London: Applied Science.
Ewing, R. (to appear), ‘Adaptive mesh refinement in reservoir simulation applications,’ to be presented at the International Conference on Accuracy Estimates and Adaptive Refinement in Finite Element Computations, Lisbon, Portugal, June, 1984.
Gresho, P.M., and Lee, R.L. (1980), “Don’t suppress the wiggles-they’re telling you something!’, in Finite Element Methods for Convection Dominated Flows, ed. by T.J.R. Hughes, New York: American Society of Mechanical Engineers, 37–61.
Heinrich, J.C., Huyakorn, P.S., Zienkiewicz, O.C., and Mitchell, A.R. (1977), “An ‘upwind’ finite element scheme for two-dimensional convective transport equation,’ Int. J. Num. Meth. Engrg. 11, 131–143.
Hughes, T.J.R. (1978), ‘A simple scheme for developing upwind finite elements,’ Int. J. Num. Meth. Engrg. 12, 1359–1365.
Huyakorn, P.S., and Pinder, G.F. (1978), “A new finite element technique for the solution of two-phase flow through porous media,’ Adv. Water Resources 1: 5, 285–298.
Jensen, O.K., and Finlayson, B.A. (1980), ’ Oscillation limits for weighted residual methods applied to convective diffusion equations,“ Int. J. Num. Meth. Engrg. 15, 1681–1689.
Mercer, J.W., and Faust, C.R. (1977), ‘The application of finite-element techniques to immiscible flow in porous media,’ presented at the First International Conference on Finite Elements in Water Resources, Princeton, Nj, July, 1976.
Nakano, Y. (1980), ‘Application of recent results in functional analysis to the problem of wetting fronts,’ Water Res. Research 16, 314–318.
Oleinik, O.A. (1963), ‘Construction of a generalized solution of the Cauchy problem for a quasi-linear equation of the first order by the introduction of a ‘vanishing viscosity’,“ Am. Math. Soc. Trans!. (Ser. 2), 33, 277–283.
Prenter, P.M. (1975), Splines and Variational Methods, New York: Wiley.
Shapiro, A.M., and Pinder, C.F. (1981), “Analysis of an upstream weighted approximation to the transport equation,’ J. Comp. Phys. 39, 46–71.
Shapiro, A.M., and Pinder, G.F. (1982), “Solution of immiscible displacement in porous media using the collocation finite element method,’ presented at the Fourth International Conference on Finite Elements in Water Resources, Hannover, FRG, June, 1982.
Von Neumann, j., and Richtmyer, R.D. (1950), “A method for the numerical calculations of hydrodynamical shocks,” J. Appl. Phys. 21, 232–237.
Whitman, G.B. (1974), Linear and Nonlinear Waves, New York: Wiley.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Allen, M.B. (1984). Why Upwinding is Reasonable. In: Laible, J.P., Brebbia, C.A., Gray, W., Pinder, G. (eds) Finite Elements in Water Resources. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11744-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-662-11744-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-11746-0
Online ISBN: 978-3-662-11744-6
eBook Packages: Springer Book Archive