Author Archives: Sophia Horak

Thesis Snapshot: ODIL in Julia (Bachelor thesis by Luca Diederich)

ODIL is a numerical method for solving partial differential equations developed and published in 2023 [1]. Instead of relying on neural networks, ODIL employs conventional grid-based PDE discretizations and optimizes a discrete loss function using gradient-based methods as well as Gauss–Newton techniques, thereby combining numerical methods with machine learning into a powerful tool for challenging inverse problems in the natural sciences. As part of this work, the approach originally developed in Python was implemented and evaluated in Julia.

The working principle of ODIL can be seen in the following picture. Burger’s equation is solved on a one dimensional domain in time and space. Sparse measurements of a precomputed Finite Volume solution (left) are prescribed at the depicted points (right, black dots on rectangle), and ODIL is run to reconstruct the solution. After 75 Gauß-Newton steps a stable solution (right) is obtained.

References:
[1] Karnakov, P., Litvinov, S. & Koumoutsakos, P. Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. Eur. Phys. J. E 46, 59 (2023). https://doi.org/10.1140/epje/s10189-023-00313-7

Workshop: Trixi User and Developer Interaction (TRUDI) 2026

From 16 to 20 March, we are hosting the TRUDI workshop at the Division of Mathematics at the University of Cologne.

TRUDI 2026 is a workshop bringing together users and developers of the Trixi Framework, which includes the Julia packages Trixi.jl, TrixiShallowWater.jl, TrixiAtmo.jl, and TrixiParticles.jl. The program will feature talks, tutorials, and hackathons aimed at consolidating the existing community, aligning long-term strategy, and advancing key features in high-performance scientific computing.

In addition to the main Trixi Framework packages, we rely on – and actively collaborate with – a broader ecosystem of upstream dependencies and partner projects maintained by members of our community, including (but not limited to):

The event is co-hosted by SNuBIC, the DFG research unit on “Structure-preserving Numerical Methods for Bulk and Interface-Coupling of Heterogeneous Models” (FOR5409). We gratefully acknowledge further support by the key profile area Dynamics of the Universe of the University of Cologne.

Further information can be found here: https://trixi-framework.github.io/trudi-2026/.

New preprint published: SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus

We extend the framework of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) operators to derive an energy-conservative and divergence-free scheme for the homogeneous Maxwell’s equations.

https://arxiv.org/abs/2511.20529v1

Abstract
We demonstrate that we can carry over the strategy of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) methods to achieve divergence- and curl-free discretizations. This is not obvious at first sight, as for SBP-FD no basis functions are known, but only values and derivatives at points. The key is a remarkable analytic relationship that enables us to construct compatible operators using integral and nodal degrees of freedom. Pre-existing SBP-FD matrix operators can then be used to obtain nodal values from the integral degrees of freedom to derive a scheme with the desired properties.

Snapshot: Outreach visit to Kreuzgasse Gymnasium Cologne

On Thursday, 15 January 2026, Simone Chiocchetti visited students of the 11. Klasse at Kreuzgasse Gymnasium in Cologne for an interactive meeting on computational physics, engineering, and applied mathematics, with a particular focus on the numerical simulation of fluids.
The discussion covered a wide range of topics: from the mathematical foundations of partial differential equations and their discretization, to real-world applications in civil engineering, astrophysics, and industrial design. Students learned how researchers model phenomena ranging from river floods and debris flows to turbulence in aerodynamics, and why such simulations matter when experiments are expensive, dangerous, or simply impossible. The conversation also touched on the practical challenges of supercomputing, multiphase flows one can observe in a cup of coffee, and the surprising behavior of steel structures under large deformations.
The students participated actively throughout, contributing thoughtful questions on subjects including chaos in planetary orbits and its connection to fluid turbulence, the multiscale nature of turbulent flows, and the computational tricks needed to make large simulations feasible.
Many thanks to the students for their engagement and curiosity, and to Markus Klein, their physics teacher, for contributing to organizing the event.
This activity is funded by the European Union’s Horizon Europe Research and Innovation Programme under the Marie Skłodowska-Curie Postdoctoral Fellowship MoMeNTUM (grant agreement No. 101109532).

Talk: Alex Bercik (University of Toronto) with Volume Dissipation for Finite-Difference and Spectral-Element Summation-By-Parts Methods

We construct provably stable, conservative, and accurate volume dissipation within a tensor-product framework. The dissipation operators can be applied to any scheme that uses the SBP framework, including high-order entropy-stable schemes. We clarify the incorporation of a variable coefficient within the operator structure, and connect the presented volume dissipation to the use of upwind SBP operators, both in a finite-difference and spectral-element context. Numerical examples featuring the linear convection, Burgers, and Euler equations verify the properties of the constructed dissipation operators, and assess their performance compared to existing upwind schemes, including local linear stability behaviour. When applied to entropy-stable schemes, the presented approach results in accurate and robust methods that can solve a broader range of problems where comparable existing methods fail.

Link to the paper: arXiv:2503.12670

Snapshot: DE Mach 2000 astrophysical Jet with DG and subcell positivity limiters using AMR

Numerical simulation of an astrophysical Jet with a Mach number of about 2000. We solve the compressible Euler equations of gas dynamics with a FCT-type subcell limiting approach combining a (forth-order accurate) discontinuous Galerkin (DG) method with a first-order accurate finite volume (FV) method at the node level to impose positivity of density and pressure. Additionally, a new mortar approach also ensures positivity of density and pressure.

At end time T=0.0015, we have 79,708 elements which result in 1,275,328 degrees of freedom. We use the entropy-conserving and kinetic energy preserving flux of Chandrashekar for the volume fluxes and the local Lax-Friedrichs flux for the surface fluxes of the DG and FV methods.

The results were obtained using Trixi.jl.

Reference:
[1] A. M. Rueda-Ramírez, B. Bolm, D. Kuzmin, G. J. Gassner: Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods, https://link.springer.com/article/10.1007/s42967-023-00321-6

Snapshot: Modern transfer learning architectures improve archaeological artifact classification

With the increasing complexity of deep neural networks and continual architectural improvements, AI models have achieved remarkable success in image classification, surpassing 90% TOP-1 accuracy on ImageNet. Such high performance highlights their effectiveness across diverse domains and supports reliable transfer learning for smaller, specialized datasets. Artifact classification is one example where knowledge transfer from large-scale datasets proves highly beneficial. In (a), a pretrained CNN from ImageNet provides powerful feature extraction for archaeological image classification. In (b), this transferred knowledge is further extended to multi-image classification tasks through optimized feature fusion, enhancing overall model performance.

Snapshot: Subcell IDP/FCT Limiting Simulation of the Mach 2000 Astrophysical Jet

The videos show a result of simulations with DGSEM and a polynomial degree of 3. At the end of the simulation, the solution has about 150000 degrees of freedom. Subcell IDP/FCT-type Limiting provides for stability. Moreover, the simulation uses adaptive mesh refinement, combined with a new IDP mortar type that ensures positivity of density and pressure.
On the left side, local (shock-capturing) limiters are used in the volume integral, while on the right side only limiters that ensure physics admissibility (positivity of density and pressure) are enabled.

Snapshot: Investigating Atmospheric Dynamics: The Held-Suarez Test Case with TrixiAtmo.jl

Our research group is currently utilizing TrixiAtmo.jl to explore the Held-Suarez test case, a widely recognized benchmark for atmospheric general circulation and climate models. First proposed by Held and Suarez in their 1994 paper, this idealized setup is designed to capture fundamental large-scale flow features of Earth’s atmosphere.

The model incorporates a simplified atmospheric forcing that establishes a decreasing temperature gradient from the equator to the poles and maintains a vertically balanced state. It also accounts for idealized boundary layer friction affecting the wind field. This configuration drives a characteristic atmospheric circulation: warm air rises at the equator, while colder air flows towards it at lower levels. The Coriolis force deflects this moving air, leading to the development of prominent jet streams in both the Northern and Southern Hemispheres.

For our simulations, TrixiAtmo.jl employs a Discontinuous Galerkin Spectral Element Method (DGSEM) with flux differencing. We discretize the Earth’s atmosphere using a cubed sphere grid with 6 patches, each consisting of 10 x 10 x 8 cells. Starting from a state with constant temperature and without any motion, the video visualizes the simulated near-surface air temperature and the developing flow patterns.