We consider a cross-diffusion system describing the transport of ions through
nanopores. For this system, existence and uniqueness have been established in
the literature under certain conditions using the entropy structure of the
system.
We use a cell-centered finite volume method to obtain a numerical
solution. We bound the difference between the exact solution and a Morley-type
reconstruction of the numerical solution by a fully computable a-posteriori
error estimator. Instead of relying on the entropy structure, we use the L^2
framework to obtain a stability estimate and later on the a-posteriori error
estimator. This is in contrast to the analytical results, which explicitly
avoid the L^2 norm, but enables us to consider arbitrarily small ion
concentrations.

Structure-preserving numerical methods for dispersive wave equations:

Several water wave propagation problems can be modeled using a
depth-averaged shallow water approximation, e.g., tsunami propagation
or dam breaks. In  many cases, the classical first-order hyperbolic
shallow water equations are sufficient to describe the wave dynamics.
However, in some cases, higher-order effects need to be taken into
account, leading to nonlinear dispersive wave equations. Several
variants of such models exist and are used in practice. In this talk,
we will review some recent developments of structure-preserving numerical
methods. In particular, we will consider invariants such as the total
energy and study efficient numerical methods yielding qualitative and
quantitative improvements compared to standard schemes.
To develop structure-preserving schemes, we make use of the general
framework of summation-by-parts (SBP) operators in space, unifying the
analysis of finite difference, finite volume, finite element,
discontinuous Galerkin, and spectral methods. Finally, we combine
structure-preserving spatial discretizations with relaxation methods in
time to obtain fully-discrete, energy-conservative schemes.

Alena Ulke (University of Mannheim):

Gas Flow of Mixtures on Networks: Existence and Uniqueness of Solutions

Climate change compels for a transition away from fossil fuels, positioning
hydrogen as a promising alternative, with hydrogen-natural gas mixtures being
a viable, interim strategy to reduce greenhouse gas emissions. Although this
leads to a growing interest in modeling and optimizing the gas flow of
mixtures
on networks, only limited theoretical results are available.

In this talk, we present a mathematical framework to describe the gas flow of
mixtures on networks. The model is based on the physical principles of the
isothermal Euler equations and the mixing of incoming flow at nodes. We then
prove the existence (and uniqueness) of solutions to the steady-state model
on tree-shaped networks and networks containing a cycle. We also underline
our theoretical results through numerical experiments and illustrate the
applicability of our proof to networks with more complex topologies, such as
those containing multiple cycles.

This talk is based on a joint work with  Simone Göttlich (University of
Mannheim) and  Michael Schuster (FAU Erlangen).

Link to preprint: https://arxiv.org/abs/2411.03841

Albert-Schweitzer-Straße 19

55128 Mainz