Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-11T09:38:05.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Discretization methods with analytical characteristicmethods for advection-diffusion-reaction equationsand 2d applications

Published online by Cambridge University Press:  01 August 2009

Jürgen Geiser
Jürgen Geiser*
Affiliation:
Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. geiser@mathematik.hu-berlin.de
Get access

Abstract

Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods aredeveloped for solving the arising systems of convection-diffusion-dispersion-reactionequations, and the received results of several discretizationmethods are presented. We concentrate on linear reaction systems, which can be solved analytically.In the numerical methods, we use large time-steps to achievelong simulation times of about 10 000 years.We propose higher-order discretization methods, which allow us to use large time-steps without losing accuracy.By decoupling of a multi-physical and multi-dimensional equation,simpler physical and one-dimensional equations are obtained and can be discretized with higher-order methods. The results can then be coupled with an operator-splitting method.We discuss benchmark problems given in the literature and real-life applications.We simulate a radioactive waste disposals with underlying flowing groundwater.The transport and reaction simulations for the decay chains are presentedin 2d realistic domains, and we discuss the received results.Finally, we present our conclusions and ideas for further works.

Keywords

Advection-diffusion-reaction equationembedded analytical solutionsoperator-splitting methodscharacteristicmethodsfinite-volume methodsmulti-physicssimulation of radioactive waste disposals
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1157 - 1183
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

T. Barth and M. Ohlberger, Finite volume methods: foundation and analysis, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes Eds., John Wiley & Sons, Ltd (2004).
Bastian, P. and Lang, S., Couplex benchmark computations with UG. Computat. Geosci. 8 (2004) 125–147. CrossRef
J. Bear, Dynamics of fluids in porous media. American Elsevier, New York, USA (1972).
J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht, Boston, London (1991).
Bourgeat, A., Kern, M., Schumacher, S. and Talandier, J., The COUPLEX test cases: Nuclear waste disposal simulation: Simulation of transport around a nuclear waste disposal site. Computat. Geosci. 8 (2004) 83–98. CrossRef
Celia, M.A., Russell, T.F., Herrera, I. and Ewing, R.E., Eulerian-Lagrangian, An localized adjoint method for the advection-diffusion equation. Adv. Wat. Res. 13 (1990) 187–206. CrossRef
Eykolt, G.R., Analytical solution for networks of irreversible first-order reactions. Wat. Res. 33 (1999) 814–826.
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis 7, Amsterdam, North Holland (2000) 713–1020.
R. Eymard, T. Gallouët and R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient, in Handbook of Numerical Analysis 37, Appl. Numer. Math. (2001) 31–53.
Eymard, R., Gallouët, T. and Herbin, R., Error estimates for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differ. Equ. 7 (2002) 419–440.
Farago, I. and Geiser, J., Iterative operator-splitting methods for linear problems. International J. Computat. Sci. Eng. 3 (2007) 255–263. CrossRef
E. Fein, Test-example for a waste-disposal and parameters for a decay-chain. Private communications, Braunschweig, Germany (2000).
E. Fein, Physical Model and Mathematical Description. Private communications, Braunschweig, Germany (2001).
E. Fein, T. Kühle and U. Noseck, Development of a software-package for three dimensional models to simulate contaminated transport problems. Technical Concepts, Braunschweig, Germany (2001).
Frolkovič, P., Flux-based method of characteristics for contaminant transport in flowing groundwater. Comput. Vis. Sci. 5 (2002) 73–83.
Frolkovič, P. and De Schepper, H., Numerical modeling of convection dominated transport coupled with density driven flow in porous media. Adv. Wat. Res. 24 (2001) 63–72. CrossRef
P. Frolkovič and J. Geiser, Numerical Simulation of Radionuclides Transport in Double Porosity Media with Sorption, in Proceedings of Algorithmy 2000, Conference of Scientific Computing (2000) 28–36.
J. Geiser, Gekoppelte Diskretisierungsverfahren für Systeme von Konvektions-Dispersions-Diffusions-Reaktionsgleichungen. Doktor-Arbeit, Universität Heidelberg, Germany (2004).
Genuchten, M.T., Convective-dispersive transport of solutes involved in sequential first-order decay reactions. Comput. Geosci. 11 (1985) 129–147.
Godunov, S.K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271–290.
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer Verlag Berlin, Heidelberg, New York (2002).
Harten, A., Enguist, B., Osher, S. and Charkravarthy, S., Uniformly high order esssentially non-oscillatory schemes I. SIAM J. Numer. Anal. 24 (1987) 279–309. CrossRef
Harten, A., Enguist, B., Osher, S. and Charkravarthy, S., Uniformly high order esssentially non-oscillatory schemes III. J. Computat. Phys. 71 (1987) 231–303. CrossRef
W.H. Hundsdorfer, Numerical Solution of Advection-Diffusion-Reaction Equations. Technical Report NM-N9603, CWI (1996).
W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics 33. Springer Verlag (2003).
Liu, X.-D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115 (1994) 200–212. CrossRef
R.J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhäuser Verlag Basel, Boston, Berlin (1992).
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002).
McLachlan, R.I., Quispel, R.G.W., Splitting methods. Acta Numer. 11 (2002) 341–434.
Morton, K.W., On the analysis of finite volume methods for evolutionary problems. SIAM J. Numer. Anal. 35 (1998) 2195–2222. CrossRef
Roache, P.J., A flux-based modified method of characteristics. Int. J. Numer. Methods Fluids 12 (1992) 1259–1275. CrossRef
Scheidegger, A.E., General theory of dispersion in porous media. J. Geophysical Research 66 (1961) 32–73.
Shu, C.-W., High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Internat. J. Comput. Fluid Dynamics 17 (2003) 107–118. CrossRef
Sonar, T., On the design of an upwind scheme for compressible flow on general triangulation. Numer. Anal. 4 (1993) 135–148.
Sportisse, B., An analysis of operator-splitting techniques in the stiff case. J. Comput. Phys. 161 (2000) 140–168. CrossRef
Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. CrossRef
Sun, Y., Petersen, J.N. and Clement, T.P., Analytical solutions for multiple species reactive transport in multiple dimensions. J. Contam. Hydrol. 35 (1999) 429–440. CrossRef
V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054. Springer Verlag, Berlin, Heidelberg (1984).
J.G. Verwer and B. Sportisse, A note on operator-splitting in a stiff linear case. MAS-R9830, ISSN (1998) 1386–3703.
Z. Zlatev, Computer Treatment of Large Air Pollution Models. Kluwer Academic Publishers (1995).