# Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 6, page 1157-1183
- ISSN: 0764-583X

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topGeiser, Jürgen. "Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1157-1183. <http://eudml.org/doc/250617>.

@article{Geiser2009,

abstract = {
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 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 presented
in 2d realistic domains, and we discuss the received results.
Finally, we present our conclusions and ideas for further works.
},

author = {Geiser, Jürgen},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Advection-diffusion-reaction equation;
embedded analytical solutions; operator-splitting methods; characteristic
methods; finite-volume methods; multi-physics;
simulation of radioactive waste disposals; advection-diffusion-reaction equation; embedded analytical solutions; characteristic methods; simulation of radioactive waste disposals},

language = {eng},

month = {8},

number = {6},

pages = {1157-1183},

publisher = {EDP Sciences},

title = {Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications},

url = {http://eudml.org/doc/250617},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Geiser, Jürgen

TI - Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/8//

PB - EDP Sciences

VL - 43

IS - 6

SP - 1157

EP - 1183

AB -
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 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 presented
in 2d realistic domains, and we discuss the received results.
Finally, we present our conclusions and ideas for further works.

LA - eng

KW - Advection-diffusion-reaction equation;
embedded analytical solutions; operator-splitting methods; characteristic
methods; finite-volume methods; multi-physics;
simulation of radioactive waste disposals; advection-diffusion-reaction equation; embedded analytical solutions; characteristic methods; simulation of radioactive waste disposals

UR - http://eudml.org/doc/250617

ER -

## References

top- 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).
- P. Bastian and S. Lang, Couplex benchmark computations with UG. Computat. Geosci.8 (2004) 125–147. Zbl1060.86001
- J. Bear, Dynamics of fluids in porous media. American Elsevier, New York, USA (1972). Zbl1191.76001
- J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht, Boston, London (1991). Zbl0780.76002
- 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. Zbl1060.86002
- 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.
- G.R. Eykolt, 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 Analysis7, Amsterdam, North Holland (2000) 713–1020. Zbl0981.65095
- R. Eymard, T. Gallouët and R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient, in Handbook of Numerical Analysis37, Appl. Numer. Math. (2001) 31–53. Zbl0982.65122
- 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. Zbl1173.35563
- I. Farago and J. Geiser, Iterative operator-splitting methods for linear problems. International J. Computat. Sci. Eng.3 (2007) 255–263.
- 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).
- P. Frolkovič, Flux-based method of characteristics for contaminant transport in flowing groundwater. Comput. Vis. Sci.5 (2002) 73–83. Zbl1052.76578
- 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.
- 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. Zbl1057.76596
- J. Geiser, Gekoppelte Diskretisierungsverfahren für Systeme von Konvektions-Dispersions-Diffusions-Reaktionsgleichungen. Doktor-Arbeit, Universität Heidelberg, Germany (2004).
- M.T. Genuchten, Convective-dispersive transport of solutes involved in sequential first-order decay reactions. Comput. Geosci.11 (1985) 129–147.
- 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. Zbl0171.46204
- E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics31. Springer Verlag Berlin, Heidelberg, New York (2002). Zbl0994.65135
- 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.
- A. Harten, B. Enguist, S. Osher and S. Charkravarthy, Uniformly high order esssentially non-oscillatory schemes III. J. Computat. Phys.71 (1987) 231–303.
- 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 Mathematics33. Springer Verlag (2003). Zbl1030.65100
- X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys.115 (1994) 200–212. Zbl0811.65076
- R.J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics. Birkhäuser Verlag Basel, Boston, Berlin (1992). Zbl0847.65053
- R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002).
- R.I. McLachlan, R.G.W. Quispel, Splitting methods. Acta Numer.11 (2002) 341–434. Zbl1105.65341
- K.W. Morton, On the analysis of finite volume methods for evolutionary problems. SIAM J. Numer. Anal.35 (1998) 2195–2222. Zbl0927.65119
- P.J. Roache, A flux-based modified method of characteristics. Int. J. Numer. Methods Fluids12 (1992) 1259–1275. Zbl0765.76057
- A.E. Scheidegger, General theory of dispersion in porous media. J. Geophysical Research66 (1961) 32–73.
- C.-W. Shu, High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Internat. J. Comput. Fluid Dynamics17 (2003) 107–118. Zbl1034.76044
- T. Sonar, On the design of an upwind scheme for compressible flow on general triangulation. Numer. Anal.4 (1993) 135–148. Zbl0765.76066
- B. Sportisse, An analysis of operator-splitting techniques in the stiff case. J. Comput. Phys.161 (2000) 140–168. Zbl0953.65062
- G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal.5 (1968) 506–517. Zbl0184.38503
- 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.
- V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics1054. Springer Verlag, Berlin, Heidelberg (1984). Zbl0528.65052
- 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).

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