Abstract
Partial differential equations with a near-frontal or frontal behaviour occur in a number of practical situations. Notable examples are the diffusion-convection and the Buckley-Leverett equations. The diffusion convection equation is a second order equation that includes a first order term. The dominance of the first order term renders the equation to a hyperbolic form exhibiting a frontal character. The Buckley-Leverett equation is a first order non-linear equation characterised with a frontal behaviour.
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., and Pinder, G.F., “Collocation Simulation of Multiphase Porous-Medium Flow”, Soc. Pet. Eng. J., Feb.1983, pp. 135–142.
Aziz, K. and Settari, A., “Petroleum Reservoir Simulation” Applied Science Publishers, Ltd. London (1979) Sec. 5. 5.
Christie, I., Griffiths, D.F., Mitchell, A.R. & Zienkiewicz, 0.C., “Finite Element Methods for Second Order Differential Equations with Significant First Derivatives”, Int.J. Num. Meth. Engng., (1976), 10, pp. 1389–1396.
Culham, W.E. & Varga, R.S. “Numerical Methods for Time Dependent, Non-linear Boundary Value Problems”, Soc. Pet. Eng. J (1971), pp. 374–388.
Dalen, V. “Simplified Finite Element Models for Reservoir Flow Problems”, Soc. Pet. Eng. J. (Oct.1979) 333–43.
Gladwell, I. and Wait, R., “A Survey of Numerical Methods for Partial Differential Eqauations”, Oxford University Press, 1979, pp. 195–211.
Grobner, W. and Hofreiter, N. “Integraltafel”, Springer, Vienna, Vol.1, 1949, Vol. 2, 1950.
Guymon, G.L., “A Finite Element Solution of the One-Dimensional Diffusion-Convection Equation”, Water Resources Res., (1970) 6, pp 204–210.
Heinrich, J.C., Huyakorn, P.S. & Zienkiewicz, O.C., “An ‘Upwind’ Finite Element Scheme for Two-Dimensional Convective Transport Equations”, Int. J. Num. Meth. Engng., (1977), 11, pp.131–143:
Mohsen, M.F.N. & Pinder, G.F., “Analytical Solution of the Transport Equation Using Polynomial Initial Condition for Verification of Numerical Simulators”. To appear in Int. J. of Num. Meth. in Fluids (1983).
Peaceman, D.W. & Rachford, H.H. Jr., “Numerical Calculation of Multi-dimensional Miscible Displacement”, Soc. Pet. Eng. J., (1962), 237, pp 327–338.
Price, H.S., Cavendish, J.C., & Varga, R.S., “Numerical Methods of High-Order Accuracy for Diffusion-Convection Equations”, Soc. Pet. Eng. J. (1968), 243, pp 293–303.
Smith, I.M., Faraday, R.V. & O’Conner, B.A., “Raleigh-Ritz and Galerkin Finite Elements for Diffusion-Convection Problems, Water Resources Res., (1973), 9, pp. 593–606.
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
Mohsen, M.F.N. (1984). Numerical Experiments Using ‘Adaptive’ Finite Elements with Collocation. 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_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-11744-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-11746-0
Online ISBN: 978-3-662-11744-6
eBook Packages: Springer Book Archive