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Issue ESAIM: M2AN
Volume 43, Number 6, November-December 2009
Page(s) 1157 - 1183
DOI 10.1051/m2an/2009033
Published online 01 August 2009

ESAIM: M2AN 43 (2009) 1157-1183
DOI: 10.1051/m2an/2009033

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

Jürgen Geiser

Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. geiser@mathematik.hu-berlin.de

Received November 2nd, 2007. Revised October 25, 2008. Published online August 1st, 2009.

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.


Mathematics Subject Classification. 35K15, 35K57, 47F05, 65M60, 65N30.

Key words: Advection-diffusion-reaction equation, embedded analytical solutions, operator-splitting methods, characteristic methods, finite-volume methods, multi-physics, simulation of radioactive waste disposals


© EDP Sciences, SMAI 2009


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