The research group ‚Numerical Simulation’ of the Department of Mathematics and Computer Science at University of Cologne works on the development and analysis of efficient numerical methods for the simulation of problems in natural sciences and engineering.
The focus is on the development of high order methods for hyperbolic and mixed hyperbolic/parabolic partial differential equations with a special emphasis on high performance computing. Applications include e.g. the compressible Navier-Stokes equations (turbulence), the ideal and resistive Magnetohydrodynamic equations (plasma) and the Shallow Water equations (ocean surface waves).
The goal is to develop novel numerical schemes and algorithms that are effective on over 100,000 processors and beyond. For this, numerical simulation relates several fields such as numerical analysis, performance engineering, data science and physics/engineering.
Projects
DFG funded Excellence Cluster DYNAVERSE (2026 – 2032). DYNAVERSE is a new Cluster of Excellence between the Universities of Cologne and Bonn, the Forschungszentrum Jülich, the Max Planck Institute for Radio Astronomy, the German Aerospace Center and the Heidelberg Institute for Theoretical Studies. It is a world-class hub of expertise in radio astronomy, lab-experiments, simulations, and machine learning / artificial intelligence. Dynaverse brings together researchers from Astrophysics, Computer Science and Mathematics. State-of-the-art research facilities to which our team has priority access combined with the expertise in high-performance computing, and data-analytics will generate new ground-breaking opportunities.
Visit the Dynaverse Excellence Cluster webpage for all information (https://dynaverse.astro.uni-koeln.de/)
BMFTR funded project ADAPTEX: ADAPtive Earth system modelling with strongly reduced computation time for EXascale-supercomputers (2023-2025). The aim of project ADAPTEX is to develop a generally usable open-source software framework for exascale-capable flow simulations on dynamic adaptive grids and its application in Earth System Modeling (ESM). By merging existing, individually specialized high-performance computing software libraries and extending them to heterogeneous exascale computer architectures, the parallel scalability of current and future CFD applications is improved, the modeled spatial resolution and accuracy increased, and overall resource efficiency improved.Our research group is concerned with extending our flow solver Trixi.jl to allow for global circulation simulations. Furthermore, we develop an interface library to make our Julia code accessible for legacy applications. We are especially interested in the application of the Discontinuous Galerkin (DG) Method and Adaptive Mesh Refinement (AMR) in the context of ESM.(Funded by the Federal Ministry of Research, Technology, and Space (BMFTR) under the funding code 16ME0668K as part of the SCALEXA initiative, and thereby also through the European Union – NextGenerationEU.)
Link to the ADAPTEX project page: https://dlr-amr.github.io/adaptex/
Advanced Meshfree RBF Methods for Hyperbolic PDEs: Application to the Shallow Water Equations in Two Dimensions and on the Sphere (2024 – ). This research focuses on the development of advanced meshfree Radial Basis Function (RBF) methods for solving hyperbolic partial differential equations (PDEs), with a particular emphasis on the shallow water equations in both two-dimensional planar and spherical geometries. The RBF framework offers remarkable flexibility and efficiency for handling complex geometries without the need for mesh generation, making it highly suitable for geophysical and environmental flow simulations. Despite its advantages, the method faces several challenges—most notably, the accurate computation of steep gradients, the handling of shocks, and the preservation of entropy. The primary objective of this work is to address these challenges and to design robust, entropy-stable RBF-Finite Difference (RBF-FD) schemes. The research aims to contribute novel formulations within various RBF frameworks, including collocation and oversampled approaches, to enhance the stability and physical fidelity of RBF-based solvers for hyperbolic systems. Furthermore, extending these methods to multilayer shallow water systems over complex domains introduces additional numerical and theoretical difficulties. This study will develop efficient and stable RBF-FD schemes for such systems while ensuring entropy consistency and robustness across layers. In addition, the application of RBF-FD to the shallow water equations on spherical coordinate systems presents unique challenges related to interpolation accuracy and gradient computation. The research will therefore focus on improving the numerical scheme to handle spherical geometries efficiently and accurately, enabling high-fidelity simulations of large-scale geophysical flows.
(This project is supported by the Hilde Domin Program of the German Academic Exchange Service (DAAD). For more information, see: https://www.daad.de/en/the-daad/hilde-domin-program/)
BMFTR funded Initiative WarmWorlds – Module Smarter, Project ICON-DG (2025 – 2027). In recent decades, advances in computational performance and model resolution have significantly enhanced climate modelling. Nevertheless, to accurately predict regional climate variations and, in particular, the occurrence and intensity of extreme weather events remains a challenge.Our aim with this project is to incorporate latest advancements in discontinuous Galerkin spectral element methods (DGSEM) into the ICON-DG weather model.Specifically, we seek to implement entropy stable and horizontally explicit vertically implicit (HEVI) schemes on prismatic elements to ensure stability for high order solvers even for large time steps.
Link to warmworlds project page https://warmworld.de/
DFG funded project “High-Order Methods for Liquid Column Chromatography with Guaranteed Preservation of Solution Bounds” (2025 – 2027). This project transfers recent advances in structure-preserving high-order discontinuous Galerkin (DG) methods to the field of liquid column chromatography—a key process for the separation and purification of biomolecules in biotechnology. The design and optimization of chromatographic processes rely on thousands of fast and reliable simulations of stiff mathematical models that must strictly preserve physical properties such as positivity. The goal of this project is to develop mathematically provable, bound-preserving numerical schemes that ensure physically consistent solutions while maintaining high-order accuracy (where possible), enabling more robust and computationally efficient simulation. These methods will be applied to a broad range of chromatography models, including the two-dimensional general rate model. Ultimately, the resulting methods and models will be integrated into the open-source simulation framework CADET, providing the chromatography community with state-of-the-art numerical tools.
Link to the DFG project page https://gepris.dfg.de/gepris/projekt/548805630
Marie Skłodowska-Curie Postdoctoral Fellowship MoMeNTUM (European Union’s Horizon Europe Research and Innovation Programme, grant agreement No. 101109532, 2024-2025).MoMeNTUM (Modern high order numerical Methods based on No-compromise moving Voronoi Tessellations, a Unified solver for continuum Mechanics) concerns the development of high order numerical methods on Voronoi grids, together with new meshing algorithms that allow to fully take advantage of the versatility provided by unstructured moving polygonal meshes, with special attention to the computational efficiency of the implementation. The project ultimately aims at building a versatile solver that can handle solid and fluid media with arbitrary rheology. This project recieves funding from the European Union’s Horizon Europe Research and Innovation Programme, under grant agreement No. 101109532.
Ministry of Culture and Science NRW funded project “HESCOR – Cultural Evolution in Changing Climate: Human & Earth System Coupled Research” (2023 – 2026). The project, part of the “Profile Building 2022” initiative by North Rhine-Westphalia’s Ministry for Culture and Science, aims to develop a new field of Human and Earth System Coupled Research. This interdisciplinary effort focuses on understanding how interactions between human and Earth systems have influenced human cultural evolution.
Please visit the project webpage for all the details: HESCOR (https://hescor.uni-koeln.de/).
DFG funded Research Unit “SNUBIC: Structure-Preserving Numerical Methods for Bulk- and Interface Coupling of Heterogeneous Models” (2022 – 2026). Funded by the German Research Foundation under the grant number DFG-FOR5409 to investigate the modeling and simulation of coupled systems described by partial differential equations (PDEs).
Please visit the research unit webpage for all the details: SNUBIC.
Klaus-Tschira-Stiftung funded Project “HiFiLab: A High-Fidelity Laboratory for the Simulation of Celestial Bodies with their Space Environment” (2022 – 2025). In this project, we focus on generating a novel computational simulation framework to describe the interaction of plasma with celestial bodies. Understanding the interaction of celestial bodies with their space environment is very important, as it often reveals information about their inner structure and the existence/composition of their atmospheres. Of fundamental importance is the question about liquid water under the icy surface of some moons of the solar system, as water is considered to be one of the essential ingredients for life as we know it.In the last years, we have successfully designed a high-order accurate 3D unstructured discontinuous Galerkin (DG) open source solver with fully parallel adaptive mesh refinement for single-fluid magnetohydrodynamics. DG methods are famous for their high accuracy, their high flexibility and extreme parallel scaling capabilities and are thus perfectly suited for complex plasma interaction simulations. We plan a major step forward regarding the physical modeling fidelity of our computational plasma framework, by extending our high-order DG solver to multi-ion MHD models that account for the interaction of electrons, ions, and neutrals. We will further apply the resulting novel computational plasma framework to simulate the Jovian moon Europa and compare our results with data taken by space missions during flybys of the moon and observations from the Hubble Space Telescope to gain insight and better understanding of the complex plasma interactions.
ERC Starting Grant “EXTREME”: An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws” (2017 – 2022). The dynamics of fluids and plasma is described by non-linear conservation laws. Transient behaviour on multiple scales involving turbulence and shocks is intrinsic to these problems. Due to their low dispersion and dissipation errors, adaptive high order numerical methods currently receive growing attention in academia and industry and form an emerging key technology. The potential benefits are massively improved computational efficiency and drastic reduction in memory consumption. Both benefits can be easily justified theoretically, in particular for a space-time-adaptive high order method. However, due to high algorithmic complexity, the theoretical performance is difficult to sustain on massively parallel supercomputers. The first challenge that we will address in this project is to design novel, exascale aware, space-time-adaptive algorithms and implement them in an open source solver that will scale on over 10^6 computing cores. Another indispensable property for the successful industrialisation of space-time-adaptive high order methods is robustness. Robustness, i.e. an “un-crashable” solver, which still retains all the positive benefits of the high order scheme is the ultimate goal of the current research on these methods. This requires new mathematical concepts. The second challenge we will address here is to construct a provable un-crashable, space-time-adaptive, high order solver without excessive artificial dissipation. Our mathematical key to achieve robustness is not intuitive at first sight: skew-symmetry. We will show that a specific skew-symmetric formulation guided by careful mathematics will allow us to design methods that are consistent with the second law of thermodynamics. This physical consistency is important as it will enable us to construct a new class of un-crashable space-time-adaptive high order methods. We will demonstrate the supremacy of this efficient and robust solver in complex large scale science and engineering applications.
This project recieves funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant agreement No. 714487.
Link to the ERC Starting Grant project page of the EU!
Project “DarkForest” (2019 – 2022) The goal of this project is to develop a scalable multiphysics framework for parallel astrophysical simulations. The (magnetized) fluid dynamics solver is based on a novel high-order DG approach with adaptive operator-based shock capturing, where a hierarchy of high-order DG operators are combined with a low-order finite volume solver in a suitable manner to capture strong shocks and turbulence. Besides self-gravity based on an octree (see project DG2RAV), an important research goal is to combine the tree-based radiation-transfer module TreeRay (Wünsch et al. in prep., Wünsch et al., 2018, MNRAS, 475, 3393) with the novel DG solver via the hierarchical mesh in a robust way to “bring light” into the currently “dark” simulation framework. The emphasis of the novel software framework is on HPC scalability with a hybrid parallelization strategy to keep the memory footprint of the TreeRay module as low as possible and to enable large scale simulations.
Klaus-Tschira-Stiftung funded Project “DG2RAV: An adaptive massively parallel
Discontinuous Galerkin solver with octree based GRAVity” (2017 – 2020). The goal of this project is to design and construct a novel framework based on a recently developed entropy stable discontinuous Galerkin (DG) discretization in combination with state-of-the-art algorithms and implementations to simulate and study the formation of multiple protostars in collapsing, magnetized, turbulent cores and the interaction of the accretion-driven protostellar jets and outflows. Due to the unprecedented fidelity and efficiency of the new simulation framework, we expect to unleash the full compute power of Tier-0 systems to substantially further our understanding of the impact of protostellar outflow feedback on the star formation process. The challenge in, and motivation for, this project is to bring together three important research disciplines: mathematics (numerics), informatics (software engineering) and natural sciences (astrophysics). Numerics: recently, major progress has been made in the development of adaptive high order DG methods for the solution of hydrodynamics (HD) and magnetohydrodynamics (MHD) problems that are provably stable, i.e. entropy stable on non-conforming grids even in the nonlinear case. These methods offer high robustness and high accuracy for problems with multiple scales in space and time, such as turbulence and shocks. Software Engineering: the software engineering aspects are of highest importance to construct a code that is capable of ’production runs’. The resulting simulation software must have excellent scaling on hundreds of thousands of processors to support the extreme resolution requirements needed when approximating problems with a massive range of scales in space and time. Astrophysics: this project combines numerics and software engineering such that the novel DG discretizations can be used to solve problems in modern theoretical astrophysics. Apart from MHD, simulations of star and protostellar disk formation feature multiple additional physical processes. Most importantly, an Octree-based gas self-gravity solver needs to be combined with the DG MHD solver.
DFG funded Project “A Unified Framework for Element-Based High-Order Summation-By-Parts Operators on Unstructured Grids” (2014 – 2017). The aim of this proposal is to construct a unified mathematical and computational framework with high order element-based summation-by-parts operators on unstructured hexahedral grids for the approximation of non-linear advection-diffusion systems. These high-order summation-by-parts operators include for instance finite differences, spectral elements, discontinuous spectral elements and finite volumes. With the summation-by-parts property, all these methods can be recast into a similar form which has to implemented only once. The implementation also includes the high-performance-computing aspects such as a unified MPI parallelisation. The proposed framework opens up a multitude of unique and novel extensions and applications such as e.g. operator-adaptation and natural all-in-one multi-physics discretisations, where in each sub-domain the operator has the same mathematical structure but is adapted locally either due to resolution/accuracy requirements or due to the underlying physical model. The project has been finalized in 2017.