# Research areas

Mathematical Fluid Mechanics

theory of hyperbolic conservation laws

- towards a proper solution concept for the multi-dim. compressible Euler equations
- convex integration, non-uniqueness for the multi-dim. compressible Euler equations

structure preserving numerical schemes for Euler and ideal magnetohydrodynamics

- well-balanced and low Mach schemes for Euler equations with gravity
- genuinely multi-dim. schemes
- structure-preserving compact high-order method method

applications in astrophysics

- numerical schemes for evolving stellar structure and evolution
- simulations of the evolution of the universe including magnetic fields

kinetic equations and plasma, theory & numerics

- multi-species models, existence and qualitative behavior
- numerical schemes for multi-species BGK equations based on a variational procedure

PDE inverse problems, kinetic models in biology with coefficients determined by experimental data

- theory: proving that it is possible to solve certain kinetic PDE inverse problems
- numerics: schemes that determine the coefficients from given experimental data

Further information on the areas of research here

I'm a PhD student working on developing and implementing numerical methods for conservation laws. My primary interests are in developing moving mesh methods for compressible flows in multiple dimensions. I also work on porous media flows in one dimension and uncertainty quantification for the same. My other interests are in writing efficient, parallelized and reproducible code for the aforementioned numerical methods.

Bacterial movement in a surrounding with an attracting chemical substance can be described on a kinetic level by the chemotaxis equation and on the macroscopic level by a Keller-Segel system. By observating the bacteria, one can gain insights in the equations and the occurring coefficients by solving the inverse problem. I compare the solutions of these inverse Problems for the kinetic chemotaxis equation and the macroscopic Keller-Segel system. This enables approximation of the kinetic solution by the macroscopic one which has the potential to speed up computations.

The main goal of my research is the numerical solution of inverse problems on a kinetic level. To this end, we want to use the interplay between kinetic and macroscopic models. Further, we want to make this process more efficient by incorporating reduced order approaches such as low-rank approximation.: The main goal of my research is the numerical solution of inverse problems on a kinetic level. To this end, we want to use the interplay between kinetic and macroscopic models. Further, we want to make this process more efficient by incorporating reduced order approaches such as low-rank approximation.

My research topic is in the area of modelling and mathematical theory for kinetic equations. I want to develop models for applications describing for example a plasma, a gas mixture with chemical reactions or the spread of aerosols. Then I underpin this models with proving in a mathematical way physical properties of the solution (entropy inequality, large-time behaviour, limit to macroscopic equations) and also mathematical properties like existence and uniqueness of solutions.

Kinetic equations describe the mesoscopic behaviour of fluids. New models are developped including more physical properties. Finally, the equations are solved numerically. I work on the numerics of a multi-species BGK-model.

My topic of research is asymptotic preserving schemes that are stationary preserving. In this project we are curious about the relationship between asymptotic preserving schemes and stationary preserving ones. For this purpose, we picked up a well-known asymptotic preserving scheme and proved it stationary preserving as well. From the way we used to prove the stationary preserving property, we are trying to find a general concept or at least cases where one can generalize that every asymptotic preserving scheme is stationary preserving.

**What I do:**

I am working on finite volume numerical methods for hyperbolic systems. Especially, I am considering compressible Euler and magnetohydrodynamics equations with gravity source term, for which I construct well-balanced methods and low Mach numerical fluxes.

**What I tell non-mathematicians when they ask what I do:**

Astrophysicists are interested in how the gas in the interior of stars are moving. For this they do computer simulations. Unfortunately, conventional simulation methods for this application tend to fail in two crucial points: Firstly, these methods are not able to even maintain the basic structure of the star, such that the gas is falling to the center of the star of moving away from it, which is not correct. Secondly, it is especially hard to simulate slow motions of the gas. I am trying to develop methods which can solve these problems.

I work on the question concerning uniqueness of solutions to the equations appearing in fluid mechanics. Here my focus lies on the compressible Euler equations, both isentropic and non-isentropic. Using the convex integration method one is able to prove that there exist infinitely many entropy solutions for particular initial data. It is also part of my research to investigate the properties of these solutions as well as the specification of the initial data leading to infinitely many solutions.

In my research I focus on the development of numerical methods to simulate fluid dynamical processes in the context of astrophysical phenomena. These include the simulation of stellar atmospheres and gas flow in glactic disks. The numerical methods used in my research include Finite-Volume and Discontinous Galerkin methods. In this context I design Well-Balanced and Asypmtotic-Preserving methods to cope with the challanges in the prescribed flow regimes. Moreover I specialize in the application of the relaxation technique to develope robust and stable schemes.