At this colloquium, we are happy to welcome:

Markus Bachmayr / Simon Boisserée (RWTH Aachen)

Analysis and numerics of nonlinear PDE systems in transient porous media flow models

In models of porous media flows, for instance in hydrogeology, the porosity of the solid matrix is typically treated as a static quantity. However, under certain circumstances, such as in soft sedimentary rocks or in magma flows, the porosity of the solid material can evolve under the influence of fluid pressure. In particular, this can lead to the formation of solitary porosity waves and of higher-porosity channels. We consider a system of nonlinear PDEs for porosity and effective pressure, based on a poroviscoelastic model, which describes such phenomena. We first discuss well-posedness of this PDE problem, which has been established in the literature only for initial porosities of high Sobolev smoothness. We present several results for porosities of low regularity, including cases with jump discontinuities that are of particular interest in geological applications. We then turn to first results on an adaptive numerical method, which is based on a fixed-point scheme inspired by the analysis, combined with a space-time least-squares formulation. This yields an appropriate treatment of discontinuities and enables spatially varying time steps, which are required for efficient approximations of the strongly spatially and temporally localized features of solutions. Partly joint work with Lisa Maria Kreusser (Bath) and Evangelos Moulas (Mainz).

Markus Schmittchen (TU Dresden)

Interacting Species Across Scales — Analysis and Applications

Travelling pulses of bacteria, pigment cells, whole patches of tissue, flocks of birds, fish schools, or even pedestrians — non-locally interacting agents are ubiquitous in nature. Often, it is possible to observe and model simple behavioural rules between two (indistinguishable) individuals based on biological or social forces. These interactions between any two individuals are typically referred to as first principles and lead to very rich and complex behaviours as soon as many individuals are involved. In the first part of the talk, we propose a non-local model for two interacting species and discuss pattern formation and phase segregation effects with applications to tissue growth and morphogenesis in zebrafish. We proceed by discussing the role of size-exclusion effects and shape on collective dynamics. We conclude the seminar by presenting formal singular limits (only hinting at the analytical intricacies) that establish bridges between different modelling paradigms of multi-agent systems.

Johannes Vrijmoed (FU Berlin)

Thermolab: A Thermodynamics Laboratory for Nonlinear Transport Processes in Open Systems

We developed a numerical thermodynamics laboratory called “Thermolab” to study the effects of the thermodynamic behavior of nonideal solution models on reactive transport processes in open systems. The equations of the state of internally consistent thermodynamic data sets are implemented in MATLAB functions and form the basis for calculating Gibbs energy. A linear algebraic approach is used in Thermolab to compute Gibbs energy of mixing for multicomponent phases to study the impact of the nonideality of solution models on transport processes. The Gibbs energies are benchmarked with experimental data, phase diagrams, and other thermodynamic software. Constrained Gibbs minimization is exemplified with MATLAB codes and iterative refinement of composition of mixtures may be used to increase precision and accuracy. All needed transport variables such as densities, phase compositions, and chemical potentials are obtained from Gibbs energy of the stable phases after the minimization in Thermolab. We demonstrate the use of precomputed local equilibrium data obtained with Thermolab in reactive transport models. In reactive fluid flow the shape and the velocity of the reaction front vary depending on the nonlinearity of the partitioning of a component in fluid and solid. We argue that nonideality of solution models has to be taken into account and further explored in reactive transport models. Thermolab Gibbs energies can be used in Cahn-Hilliard models for nonlinear diffusion and phase growth. This presents a transient process toward equilibrium and avoids computational problems arising during precomputing of equilibrium data.