| Zeit: |
28. Mai 2009 |
| Referent*in: |
Prof. Dr. Peter Knabner
Lehrstuhl für angewandte Mathematik, Universität Erlangen |
| Veranstaltungsort: |
Pfaffenwaldring 61, Raum U1.003 (MML), Universität Stuttgart
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| Download als iCal: |
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Detailed modelling of reactive transport processes in the undergroundoften requires the consideration of a wide range of reactive species. Aprominent example is natural attenuation, that is the assessment andmonitoring of microbially catalysed degradation processes of organiccontaminants in the subsoil or aquifer. The reactions exhibit a widerange of reaction times, which advises to model those reactions beingmuch faster than the time scale of the transport processes in a quasistationary manner, e.g. as (algebraicly described) equilibrium processes. Additionally not only mobile species (in solution) appear, butalso immobile ones (attached to the porous skeleton). Reactions withminerals play a special role as their equilibrium description is rather acomplementarity system than an algebraic equation with consequencesfor the kinetic description. In summary, the resulting system is notsemilinear and parabolic, but rather quasilinear and couples partialdifferential equations (pde), ordinary differential equations, algebraicequations and complementarity conditions. For such a system without growth bounds in the nonlinearity no general existence theory ispossibly, but recent results indicate that for special structures of thenonlinearities resulting e.g. from the mass action law, a global existencetheory is possible on the basis of a generalized Ljapunov approach.
We will review the current state of the analysis. Concerning the numerical simulation of such systems, an often used approach is operatorsplitting, in which transport and reaction becomes (iteratively) decoupled. This procedure either introduces a further consistency error (inthe non-interactive version) which can only be controlled by the timestepping, or applies a fixed point type iteration of unclear convergenceproperties. We rather propose, after appropriate (mixed) finite element discretization, to deal with the full discrete nonlinear system (by a damped Newton's method). To make the problem still feasible weadvise two means: The first is concerned with the continuous modeland aims at a transformation of the dependent variables such that asmany as possible are determined by decoupled linear pde's or by localalgebraic relations, leading to a smaller coupled system. The problemlies here in the combined appearance of kinetics and equilibrium andmobile and immobile species. Alternatively to this exact a priori decoupling we use an a posteriori decoupling on the level of the linear systemof equation in the Newton's method by ignoring weak couplings in theJacobian matrix. The resulting benefit in the solution of the linear system should supersede a possible deterioration in the convergence of theiterative method, being now only an approximate Newtons's method.The approaches are all illustrated with realistic problems, including theMoMas benchmark.
