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Solution of contaminant transport with adsorptionin porous media by the method of characteristics

Published online by Cambridge University Press:  15 April 2002

Jozef Kacur  and
Roger Van Keer
Jozef Kacur
Affiliation:
Faculty of Mathematics and Physics, Comenius University Mlynska Dolina, 84815 Bratislava, Slovakia. (kacur@fmph.uniba.sk)
Roger Van Keer
Affiliation:
Ghent University, Department of Mathematical Analysis, Galglaan 2, 9000 Gent, Belgium. (roger.vankeer@cage.rug.ac.be)
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Abstract

A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.

Keywords

Relaxation methodmethod of characteristicscontaminanttransportconvection-diffusion with adsorption.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 5 , September 2001 , pp. 981 - 1006
Copyright
© EDP Sciences, SMAI, 2001

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References

Alt, H.W. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341.
Barrett, J.W. and Knabner, P., Finite element approximation of transport of reactive solutes in porous media. Part i: error estimates for nonequilibrium adsorption processes. SIAM J. Numer. Anal. 34 (1997) 201-227. CrossRef
Barrett, J.W. and Knabner, P., Finite element approximation of transport of reactive solutes in porous media. Part ii: error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34 (1997) 455-479. CrossRef
Barrett, J.W and Knabner, P., An improved error bound for a Lagrange-Galerkin method for contaminant transport with non-Lipschitzian adsorption kinetics. SIAM J. Numer. Anal. 35 (1998) 1862-1882. CrossRef
J. Bear, Dynamics of Fluids in Porous Media. Elsevier, New York (1972).
Bermejo, R., Analysis of an algorithm for the Galerkin-characteristics method. Numer. Math. 60 (1991) 163-194. CrossRef
Bermejo, R., Galerkin-characteristics, A algorithm for transport-diffusion equation. SIAM J. Numer. Anal. 32 (1995) 425-455. CrossRef
Dawson, C.N., Godunov-mixed methods for advection diffusion equations in multidimensions. SIAM J. Numer. Anal. 30 (1993) 1315-1332. CrossRef
Dawson, C.N., Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35 (1998) 1709-1724. CrossRef
Dawson, C.N., van Duijn, C.J., and Grundy, R.E., Large time asymptotics in contaminant transport in porous media. SIAM J. Appl. Math. 56 (1996) 965-993. CrossRef
Dawson, C.N., van Duijn, C.J., and Wheeler, M.F., Characteristic-Galerkin methods for contaminant transport with non-equilibrium adsorption kinetics. SIAM J. Numer. Anal. 31 (1994) 982-999. CrossRef
Douglas, R and Russel, T.F., Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19 (1982) 871-885. CrossRef
N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. John Wiley & Sons Ltd., New York (1959).
Grundy, R.E., van Duijn, C.J., and Dawson, C.N., Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media. Quart. J. Mech. Appl. Math. 1 (1994) 69-106. CrossRef
Jäger, W. and Kacur, J., Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 605-627. CrossRef
Kacur, J., Solution of some free boundary problems by relaxation schemes. SIAM J. Numer. Anal. 36 (1999) 290-316. CrossRef
Kacur, J., Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal. 19 (1999) 119-154. CrossRef
Kacur, J. and Luckhaus, S., Approximation of degenerate parabolic systems by nondegenerate elliptic and parabolic systems. Appl. Numer. Math. 25 (1997) 1-21.
J. Kacur, Solution of convection-diffusion problems with the memory terms, in Applied Mathematical Analysis, A. Sequiera, H. Beirao de Veiga, and J.H. Videman, Eds., Kluwer Academic, Plenum Publ., New York (1999) 199-212.
P. Knabner, Mathematische Modelle für den Transport gelöstes Stoffe in sorbierenden Porösen Medien. Habilitationschrift, University of Augsburg, Germany (1989).
P. Knabner and F. Otto, Solute transport in porous media with equilibrium and nonequilibrium multiple site adsorption: uniqueness. To appear.
A. Kufner, O. John, and S. Fucík, Function Spaces. Noordhoff International Publishing, Leyden; Publishing House of the Czechoslovak Academy of Sciences, Prague (1977).
Morton, K.W., Priestly, A., and Suli, E., Stability of the Lagrange-Galerkin method with non-exact integration. RAIRO Modél. Math. Anal. Numér. 4 (1988) 225-250.
J. Necas, Les méthodes directes en théorie des équations elliptiques. Academia, Prague (1967).
Otto, F., L1-contraction and uniqueness for quasilinear elliptic-parabolic equations. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 105-110.
Pironneau, P., On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38 (1982) 309-332. CrossRef
van Duijn, C.J. and Knabner, P., Solute transport in porous media with equilibrium and nonequilibrium multiple site adsorption: Traveling waves. J. Reine Angew. Math. 415 (1991) 1-49.