Our research focuses on the structure-preserving high-order methods for partial and ordinary differential equations. High-order methods are very efficient for time-dependent problems dominated by transport effects because of their low numerical dispersion and dissipation errors compared to classical low-order methods. This enables efficient high-fidelity simulations of complex structures over long time spans and in complex geometries. Preserving important structures in discretizations improves the robustness of these methods and allows us to transfer results such as stability estimates available at the continuous level to the discrete level. Moreover, structure-preserving methods often have not only better qualitative but also improved quantitative properties such as a decreased error growth rate.

In addition, we are also interested in the efficient implementation of these numerical methods in easy-to-use open source software for modern hardware and applications in science and engineering. This includes computational fluid dynamics (CFD) problems such as the flow around cars or airplanes as well as problems in astro- and space physics such as plasma interactions at comets.

Below, we give a brief glimpse at some of our research topics. You can find more keywords on the main page and in our section on publications.

Entropy stable semidiscretizations of conservation laws

Modern kinetic energy preserving and entropy stable numerical methods are very robust for the simulation of compressible flows. In particular, they are able to simulate under-resolved flows and turbulence. We are interested in discontinuous Galerkin (DG) methods and related schemes using summation-by-parts (SBP) operators. This includes also continuous Galerkin methods, finite difference schemes, finite volumes, and flux reconstruction (FR) approaches.

Next to entropy stability and efficient implementations, adaptivity is a key ingredient for robust and reliable simulations. Combined with a controlled amount of artificial dissipation, this technology enables simulations of demanding applications in science and engineering.

Moreover, entropy-stable numerical methods provide promising approaches for the developing theory of dissipative weak solutions, measure-valued solutions, and statistical solutions. Since the Cauchy problem for nonlinear systems of conservation laws in multiple space dimensions is generally ill-posed in the sense of classical entropy weak solutions, these new solution concepts are promising paths of future investigations.

Selected papers

Efficient implementation and software development

Developing numerical methods is only the first step. To make them really useful for a general audience, they also need to be available in open source software. We follow the ideal of open science and open data, including publishing our preprints on and contributing to open source software. In particular, we publish our numerical methods for conservation laws in the Julia package Trixi.jl, which was also used to generate the videos shown on this website.

This interdisciplinary area of research spans mathematics, computer science, and several application sciences such as physics or aeronautical engineering. We bridge classical discipline boundaries and work together experts of all fields to advance the state of the art of simulation software. Moreover, we focus on making it easy to access and use our software, e.g., for research collaborators and also for students interested in writing theses (Bachelor, Master, PhD) or working on classroom projects.

You can also find more information about our software development interests on my GitHub page. If you are interested in any of the software projects linked there, please contact me.

Selected papers

Reliable and efficient time integration methods

Developing advanced semidiscretizations of conservation laws and implementing them efficiently on modern hardware is only one building block of efficient software for simulations. On top of that, optimized time integration methods have to be applied with robust and reliable step size control for adaptivity in time. Furthermore, we are interested in structure-preserving techniques also at the fully discrete level. For example, we have developed relaxation methods to preserve the conservation or dissipation of arbitrary functionals such as energy or entropy.

Furthermore, we are interested in structure-preserving properties of time integration methods and their interactions with structural properties of spatial semidiscretizations. This includes classical topics of geometric numerical integration but also properties such as positivity preservation, e.g., for fluid flows or in biological applications.

We are also interested in the basic analysis of properties of time integration methods. This includes for example the analysis of stability and conservation properties or software assisting in the analysis of these schemes, e.g., NodePy, and BSeries.jl.

Selected papers