Short Course "The mathematics of multicomponent reactive transport and flow models"

Nov 11 - Dec 21, 2018

University of Stuttgart, Pfaffenwaldring 57, rooms 2.136 and 8.122

Members and associated international members of SFB 1313 and IRTG DROPIT are invited to attend the short course. Participants will get 3 credits for attending the course.

This course is suited for Graduate and Ph.D. students from Mathematics, Computer Science, Natural Sciences and Engineering.

The course is free of charge. Possible costs for accommodation or travel expenses cannot be reimbursed.

In this lecture series we will address various mathematical aspects of modeling and simulating general reactive multicomponent transport problems (in porous media). The important aspect of porous media is to have not only homogeneous reactions (i.e. in one phase) as in classical chemical engineering, but heterogeneous reactions due to the involvement of several phases (i.e. liquid-solid). In a microscopic formulation theses are surface processes, in a macroscopic one they give size to species obeying different transport mechanism. “General” not only means general reaction networks, but also quasi-stationary equilibrium description besides kinetic ones and their combinations.

In the Modeling Part (app. 10 lectures) we will discuss various examples on the way to a macroscopic general formulation. This will show that additionally to the classification above also a distinction between (surface) reactions like adsorption or ion exchange and “classical” dissolution/precipitation reactions is necessary. The Linear Algebra of reaction networks gives rise to a reduced formulation, which is also numerically beneficial. Further questions are the global in time existence of solutions and the derivations of such macroscopic upscaled models by periodic homogenization. Typically a surface reaction changes the topology of the porous medium. If this cannot be ignored, problems with evolving microstructure emerge, for which classical and recent (micro-macro) models will be introduced.

In the Simulation Part (app. 5 lectures) we will concentrate on the efficient resolution of (large) sets of nonlinear equations or complementarity systems, as they arise by any discretization in time and space of the aforementioned models. There in a particular structure due to the spatial coupling for each species and the local reactive coupling between the species. There has been a long lasting discussion whether splitting type or all-in-one type methods are to be preferred. We will use the notion of inexact and approximative Newton’s methods to provide a unifying framework, in which also the (approximative) use of iterative methods to solve the set of linear equations for the (Newton) update and/or the use of approximations for the Jacobian (up to Jacobi-free methods) can be analyzed.