Author Archives: Sophia Schmickler

New preprint published: Mimetic Metrics for the DGSEM

Our new preprint is available on arXiv. We explore an alternative approach to the Kopriva metric terms, which is based on finite element exterior calculus.

https://doi.org/10.48550/arXiv.2410.14502

Abstract
Free-stream preservation is an essential property for numerical solvers on curvilinear grids. Key to this property is that the metric terms of the curvilinear mapping satisfy discrete metric identities, i.e., have zero divergence. Divergence-free metric terms are furthermore essential for entropy stability on curvilinear grids. We present a new way to compute the metric terms for discontinuous Galerkin spectral element methods (DGSEMs) that guarantees they are divergence-free. Our proposed mimetic approach uses projections that fit within the de Rham Cohomology.

Snapshot: Barotropic Instability with and without Orography

The video shows the evolution of a barotropic instability in Earth’s polar jet stream. Initially uniform, the jet stream undergoes perturbations, leading to vortex formation driven by the Coriolis force due to Earth’s rotation. To simulate this phenomenon, we discretize the Shallow Water Equations on the sphere using a discontinuous Galerkin method. We compare two scenarios: an ocean-covered Earth (without orography) and a more realistic representation that includes Earth’s orography. Orography is incorporated into the equations as a non-conservative term, with values sourced from the ETOPO dataset provided by the National Oceanic and Atmospheric Administration (NOAA). The simulations are performed using TrixiAtmo.jl.

New paper published: TrixiParticles.jl: Particle-based multiphysics simulation in Julia

Our new paper “TrixiParticles.jl: Particle-based multiphysics simulation in Julia” has been published in the Journal of Open Source Software.
We are happy we were able to contribute to the publication and thank all our collaborators for the great experience.


DOI

Summary

TrixiParticles.jl is a Julia-based open-source package for particle-based multiphysics simulations and part of the Trixi Framework. It handles complex geometries and specialized applications, such as computational fluid dynamics (CFD) and structural dynamics, by providing a versatile platform for particle-based methods. TrixiParticles.jl allows for the straightforward addition of new particle systems and their interactions, facilitating the setup of coupled multiphysics simulations such as fluid-structure interaction (FSI). Furthermore, simulations are set up directly with Julia code, simplifying the integration of custom functionalities and promoting rapid prototyping

Snapshot: Hyperbolic viscous flow – Three-dimensional cubic Lid-Driven Cavity

This research has been carried out by Simone Chiocchetti, funded by the European Union’s Horizon Europe Research and Innovation Programme under the Marie Skłodowska-Curie Postdoctoral Fellowship MoMeNTUM (grant agreement No. 101109532).

x-z central cross section of 3d Lid-Driven Cavity flow, solving the unified model of Godunov, Peshkov, and Romenski for hyperbolic viscous flow. The upper row shows the macroscopic flow state, while the bottom row shows the distortion field, tracking fluid flow deformations as if it were a solid. The numerical solver is a simple explicit second order Finite Volume method (MUSCL-Hancock) using an HLL-type Riemann solver based on Toro-Vazquez flux splitting. The Reynolds number is 1000 and the mesh resolution is 384^3.

Snapshot: Postdoctoral researcher Dr. Boqiang Huang joins our research group

Boqiang Huang received his Ph.D. in Biomedical Engineering from the Department of Electronic Engineering at Fudan University, Shanghai, China, in 2010. Following his doctoral studies, he was awarded the Alexander von Humboldt Postdoctoral Fellowship and worked under the mentorship of Prof. Angela Kunoth. During this time, he contributed as a research scientist in Prof. Kunoth’s group (AG-Kunoth) at the Institute of Mathematics, University of Paderborn, and later at the University of Cologne. In 2021, Dr. Huang joined the group of Prof. Dorit Merhof (AG-Merhof) at the Institute of Imaging and Computer Vision at RWTH Aachen University, and subsequently at the University of Regensburg. His research expertise spans multiple disciplines in the field of data science, including biomedical engineering, electronic engineering, applied mathematics and applied physics.

In July 2024, Dr. Huang became a research scientist in the group of Prof. Gregor Gassner (AG-Gassner). He is also supported by the HESCOR research project (“Human & Earth System Coupled Research” https://hescor.uni-koeln.de/). His current research focuses on the “Machine learning & Culture Clusters” over large timescales, ranging from the Paleolithic era (1.4 million to 24,000 years ago) to the present. He collaborates with researchers from archaeology, geophysics, and the humanities on this interdisciplinary project.

Snapshot: Postdoctoral researcher Dr. Tristan Montoya joins our research group


Tristan Montoya completed his PhD at the University of Toronto Institute for Aerospace Studies under the supervision of Prof. David W. Zingg. His doctoral work focused on the development and analysis of provably stable discontinuous spectral-element methods (DSEMs) with the summation-by-parts property for systems of conservation laws, particularly on triangular and tetrahedral meshes.

Tristan joined the Numerical Simulation research group at the University of Cologne in January 2024, and is part of the Trixi.jl development team. His current research involves the development of split-form and entropy-stable DSEMs for atmospheric models and their application to global weather and climate prediction. You can keep up-to-date with Tristan’s research and software contributions on his Google Scholar and GitHub pages, as well as on his personal website.

Snapshot: Discontinuous Galerkin Solver for the Spherical Shallow Water Equations

A preliminary step in the development of a global atmospheric model is the construction of a suitable horizontal discretization for partial differential equations in spherical geometry. The spherical shallow water equations, which consist of a scalar equation governing mass conservation coupled with a vector equation for the momentum balance under gravitational and Coriolis forces, serve as a simplified model for the horizontal dynamics of the Earth’s atmopheric circulation. Since the spherical shallow water equations exhibit many of the characteristic features and numerical challenges associated with atmospheric fluid flow, they provide a useful testbed for the development and assessment of numerical schemes for weather prediction and climate modelling (see, for example, Williamson et al. [1]).

By extending Trixi.jl to solve hyperbolic partial differential equations on curved manifolds, we are able to simulate shallow-water flows on the surface of the sphere using discontinuous Galerkin methods formulated with respect to the two-dimensional tangent space associated with a cubed-sphere grid [2, 3]. The video below depicts the relative vorticity field for the numerical solution to the spherical shallow water equations in flux form, which we discretize similarly to Bao et al. [4] using a discontinuous Galerkin method employing 5400 curved quadrilateral elements of polynomial degree seven. The initial condition corresponds to a mid-latitude jet with a small perturbation added to initiate a barotropic instability, which was proposed by Galewsky et al. [5] as a test case exhibiting complex nonlinear dynamics characteristic of those present in numerical weather prediction and climate models.


References

[1] D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber. A standard test set for numerical approximations to the shallow water equations in spherical geometry. Journal of Computational Physics, 102(1):211-224, 1992.

[2] R. Sadourny. Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids. Monthly Weather Review, 100(2):136-144, 1972.

[3] C. Ronchi, R. Iacono, and P. S. Paolucci. The “cubed sphere”: A new method for the solution of partial differential equations in spherical geometry. Journal of Computational Physics, 124(1):93-114, 1996.

[4] L. Bao, R. D. Nair, and H. M. Tufo. A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere. Journal of Computational Physics, 271:224-243, 2014.

[5] J. Galewsky, R. K. Scott, and L. M. Polvani. An initial-value problem for testing numerical models of the global shallow-water equations. Tellus A, 56(5):429–440, 2004.

This snapshot was created by Tristan Montoya.

Snapshot: Postdoctoral researcher Dr. Simone Chiocchetti joins our research group

Simone Chiocchetti studied Civil, Environmental, and Mechanical Engineering at the University of Trento, Italy, under the supervision of Prof. Dr.-Ing. Michael Dumbser.

He spent a year in Stuttgart at the IAG (Institute of Aerodynamics and Gasdynamics), funded by the DFG project “DROPIT startup grant” and he is currently working at the University of Cologne in the Numerical Simulation Research group led by Prof. Dr.-Ing. Gregor Gassner, under a Marie Skłodowska-Curie Postdoctoral Fellowship (European Union’s Horizon Europe Research and Innovation Programme, grant agreement No. 101109532).

Simone’s research interests include: high order numerical methods, of the Finite Volume and Discontinuous Galerkin families, in the context of first order hyperbolic partial differential equations; algorithms for generation and optimization of high quality unstructured meshes; treatment of stiff algebraic source terms; modeling of fluid and solid mechanics as well as multiphase flow; high-performance computing; semi-implicit numerical methods.

Simone’s current MSCA project (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.

Snapshot: Postdoctoral researcher Dr. Aleksey Sikstel joins our research group


Aleksey defended his doctoral thesis on coupling of hyperbolic conservation laws in 2020 at the RWTH Aachen University while working with Prof. Siegfried Müller and Prof. Michael Herty. After that he continued working with Prof. Jan Giesselmann at the TU Darmstadt on a posteriori error estimators for systems of hyperbolic conservation laws before switching to Cologne in 2022.

Alekseys research focus lies on adaptivity for high-order numerical methods for hyperbolic conservation laws, coupling of conservation laws via sharp interfaces, multiphase flows and stochastic-Galerkin PDE. In the Numerical Simulation Research Group Aleksey is working on multi-ion MHD systems and novel coupling and temporal schemes within the Snubic-project.