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

**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 – )** 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**

**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.**

*Discontinuous Galerkin solver with octree based GRAVity” (2017 – 2020)*.**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.