Geiser, Jürgen
Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) no. 6 , p. 1157-1183
Zbl pre05636850 | MR 2588436
doi : 10.1051/m2an/2009033
URL stable :

Classification:  35K15,  35K57,  47F05,  65M60,  65N30
Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convection-diffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10000 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 presented in 2d realistic domains, and we discuss the received results. Finally, we present our conclusions and ideas for further works.


[1] 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).

[2] P. Bastian and S. Lang, Couplex benchmark computations with UG. Computat. Geosci. 8 (2004) 125-147. Zbl 1060.86001

[3] J. Bear, Dynamics of fluids in porous media. American Elsevier, New York, USA (1972).

[4] J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht, Boston, London (1991). Zbl 0780.76002

[5] A. Bourgeat, M. Kern, S. Schumacher and J. Talandier, The COUPLEX test cases: Nuclear waste disposal simulation: Simulation of transport around a nuclear waste disposal site. Computat. Geosci. 8 (2004) 83-98. Zbl 1060.86002

[6] M.A. Celia, T.F. Russell, I. Herrera and R.E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation. Adv. Wat. Res. 13 (1990) 187-206.

[7] G.R. Eykolt, Analytical solution for networks of irreversible first-order reactions. Wat. Res. 33 (1999) 814-826.

[8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis 7, Amsterdam, North Holland (2000) 713-1020. MR 1804748 | Zbl 0981.65095

[9] 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. MR 1825115 | Zbl 0982.65122

[10] R. Eymard, T. Gallouët and R. Herbin, Error estimates for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differ. Equ. 7 (2002) 419-440. MR 1869118 | Zbl 1173.35563

[11] I. Farago and J. Geiser, Iterative operator-splitting methods for linear problems. International J. Computat. Sci. Eng. 3 (2007) 255-263.

[12] E. Fein, Test-example for a waste-disposal and parameters for a decay-chain. Private communications, Braunschweig, Germany (2000).

[13] E. Fein, Physical Model and Mathematical Description. Private communications, Braunschweig, Germany (2001).

[14] 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).

[15] P. Frolkovič, Flux-based method of characteristics for contaminant transport in flowing groundwater. Comput. Vis. Sci. 5 (2002) 73-83. Zbl 1052.76578

[16] P. Frolkovič and H. De Schepper, Numerical modeling of convection dominated transport coupled with density driven flow in porous media. Adv. Wat. Res. 24 (2001) 63-72.

[17] 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. Zbl 1057.76596

[18] J. Geiser, Gekoppelte Diskretisierungsverfahren für Systeme von Konvektions-Dispersions-Diffusions-Reaktionsgleichungen. Doktor-Arbeit, Universität Heidelberg, Germany (2004). Zbl 1060.65111

[19] M.T. Genuchten, Convective-dispersive transport of solutes involved in sequential first-order decay reactions. Comput. Geosci. 11 (1985) 129-147.

[20] S.K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271-290. MR 119433

[21] 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). MR 1904823 | Zbl 0994.65135

[22] A. Harten, B. Enguist, S. Osher and S. Charkravarthy, Uniformly high order esssentially non-oscillatory schemes I. SIAM J. Numer. Anal. 24 (1987) 279-309. MR 881365 | Zbl 0627.65102

[23] A. Harten, B. Enguist, S. Osher and S. Charkravarthy, Uniformly high order esssentially non-oscillatory schemes III. J. Computat. Phys. 71 (1987) 231-303. MR 897244 | Zbl 0652.65067

[24] W.H. Hundsdorfer, Numerical Solution of Advection-Diffusion-Reaction Equations. Technical Report NM-N9603, CWI (1996).

[25] W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics 33. Springer Verlag (2003). MR 2002152 | Zbl 1030.65100

[26] X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115 (1994) 200-212. MR 1300340 | Zbl 0811.65076

[27] R.J. Leveque, Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhäuser Verlag Basel, Boston, Berlin (1992). MR 1153252 | Zbl 0723.65067

[28] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002). MR 1925043 | Zbl 1010.65040

[29] R.I. Mclachlan, R.G.W. Quispel, Splitting methods. Acta Numer. 11 (2002) 341-434. MR 2009376 | Zbl 1105.65341

[30] K.W. Morton, On the analysis of finite volume methods for evolutionary problems. SIAM J. Numer. Anal. 35 (1998) 2195-2222. MR 1655843 | Zbl 0927.65119

[31] P.J. Roache, A flux-based modified method of characteristics. Int. J. Numer. Methods Fluids 12 (1992) 1259-1275. Zbl 0765.76057

[32] A.E. Scheidegger, General theory of dispersion in porous media. J. Geophysical Research 66 (1961) 32-73.

[33] C.-W. Shu, High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Internat. J. Comput. Fluid Dynamics 17 (2003) 107-118. MR 1967737 | Zbl 1034.76044

[34] T. Sonar, On the design of an upwind scheme for compressible flow on general triangulation. Numer. Anal. 4 (1993) 135-148. MR 1207092 | Zbl 0765.76066

[35] B. Sportisse, An analysis of operator-splitting techniques in the stiff case. J. Comput. Phys. 161 (2000) 140-168. MR 1762076 | Zbl 0953.65062

[36] G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506-517. MR 235754 | Zbl 0184.38503

[37] Y. Sun, J.N. Petersen and T.P. Clement, Analytical solutions for multiple species reactive transport in multiple dimensions. J. Contam. Hydrol. 35 (1999) 429-440.

[38] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054. Springer Verlag, Berlin, Heidelberg (1984). MR 744045 | Zbl 0528.65052

[39] J.G. Verwer and B. Sportisse, A note on operator-splitting in a stiff linear case. MAS-R9830, ISSN (1998) 1386-3703.

[40] Z. Zlatev, Computer Treatment of Large Air Pollution Models. Kluwer Academic Publishers (1995).