# Workshop: Novel Adaptive Discontinuous Galerkin Approaches for the Simulation of Atmospheric Flows

On November 30th and December 1st, the workshop on “Novel Adaptive Discontinuous Galerkin Approaches for the Simulation of Atmospheric Flows” will take place at the Department of Mathematics and Computer Science with researchers from the German Aerospace Center (DLR) Cologne, Deutscher Wetterdienst (DWD), and University of Cologne (UoC). This workshop is supported by and embedded into the Center for Earth System Observation and Computational Analysis (CESOC).

The idea of the workshop is to discuss novel developments for discontinuous Galerkin methods applied to atmospheric flow simulations, with a focus on implicit time integration, meshing of the sphere, adaptivity, efficient implementations, benchmark test cases for atmospheric flows, entropy-stable split formulations of the nonlinear partial differential equations, and bounds-preserving numerical schemes.

A detailed program with abstracts is available here.

# Talk: Deniz Bezgin (TUM) and Aaron Buhendwa (TUM) about Differentiable Fluid Dynamics in JAX: Challenges and Perspectives, Friday, 26th August 2022, 10am CEST

JAX-FLUIDS is a CFD solver written in Python, which uses the JAX framework to enable automatic differentiation (AD). This allows one to easily create applications for data-driven simulations or other optimization problems.The talk is based on the recent preprint “JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows” (arXiv:2203.13760).

To obtain the Zoom link for this online talk, please get in touch with Gregor Gassner or Michael Schlottke-Lakemper.

# Start of new Research Project: 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.

# Start of new Research Project: 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.

# New Open Position: PostDoc in Scientific Computing

The Numerical Simulation group of Professor Gassner invites applications for a 2 year (possibility for extensions afterwards available) postdoc position in scientific computing (pay grade 100% TVL-13).

For questions, please directly contact Professor Gassner via Email (ggassner@uni-koeln.de).

# JuliaCon 2021: Adaptive and extendable numerical simulations with Trixi.jl

Trixi.jl is a numerical simulation framework for adaptive, high-order discretizations of conservation laws. It has a modular architecture that allows users to easily extend its functionality and was designed to be useful to experienced researchers and new users alike. In this talk, we give an overview of Trixi’s current features, present a typical workflow for creating and running a simulation, and show how to add new capabilities for your own research projects.

This talk was given on July 30th, 2021 by Michael Schlottke-Lakemper and Hendrik Ranocha as part of JuliaCon 2021.

Repository: https://github.com/trixi-framework/talk-2021-juliacon
Conference agenda entry: https://live.juliacon.org/talk/VAGFD7

# Talk on 2021-07-02: On Wave Propagation Characteristics, Upwind SBP Properties and Energy Stability of DG Viscous Flux Discretizations

Speaker: Dr. Sigrun Ortleb, University of Kassel, Germany
Date & time: Friday, 2nd July 2021, 10 am (CEST)

Abstract:
Regarding accuracy and stability of numerical schemes for computational fluid dynamics, the investigation of diffusion/dispersion errors depending on the wave number is of utmost importance. Especially for high order methods, a desired small numerical dissipation competes with robustness and thus has to be carefully analyzed. This wave propagation analysis is often based on pure advection problems. In the literature, various approaches to discretize diffusion terms within a DG scheme have been introduced since the discretization of higher order spatial derivatives within the DG framework is less natural than in case of first order derivatives. In this talk, we will address significant differences in the disspation/dispersion properties for linear advection-diffusion, depending on the specific DG viscous flux discretization which is employed. In addition, results on energy stability of DG viscous flux formulations are dealt with and we show how to formulate the well-known LDG and BR1 fluxes in terms of global upwind SBP operators which complements the derivation and analysis regarding element level SBP properties of the DG scheme.

# Talk on 2021-05-21: On a linear stability issue of split form schemes for compressible flows

Speaker: Dr. Vikram Singh, Max-Planck-Institute for Meteorology
Date & time: Friday, 21st May 2021, 10 am (CEST)

Abstract:
Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic energy preservation leading to a reduction in aliasing errors at high orders. Recently, linear stability issues have been pointed out for these schemes for a density wave problem and we investigate this behaviour for some standard split forms. By deriving linearized equations of split form schemes, we show that most existing schemes do not satisfy a perturbation energy equation that holds at the continuous level. A simple modification to the energy flux of some existing schemes is shown to yield a scheme that is consistent with the energy perturbation equation. Numerical tests are given using a discontinuous Galerkin method to demonstrate these results.

Note: This meeting will be recorded.

# Talk on 2021-05-10: A two-dimensional stabilized discontinuous Galerkin method on curvilinear embedded boundary grids

Speaker: Dr. Andrew Giuliani, New York University
Date & time: Monday, 10th May 2021, 4pm (CEST)/10am (EDT)

Abstract:
In this talk, we present a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells when explicit time steppers are used. Thus, the scheme can take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. We prove that state redistribution is conservative and p-exact. Finally, we solve a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries. Numerical experiments reveal that our scheme converges with order $p+1$ in $L_1$ and between $p$ and $p+1$ in $L_\infty$.

# Talk on 2021-03-18: A tour of BifurcationKit and some results on mean fields of spiking neurons

Speaker: Dr. Romain Veltz, INRIA, France
Date: Thursday, 18th March 2021, 10am (CET)

## Abstract

In this talk, I will first present the basics of bifurcation theory. Then, I will give a panorama of BifurcationKit.jl, a Julia package to perform numerical bifurcation analysis of large dimensional equations (PDE, nonlocal equations, etc) possibly on GPUs using Matrix-Free / Sparse Matrix formulations of the problem. Julia programming language gives access to a rich ecosystem (PDE, GPU, AD, cluster…). Notably, numerical bifurcation analysis can be done entirely on GPU as will be shown in an example.

BifurcationKit incorporates continuation algorithms (PALC, deflated continuation, …) which can be used to perform fully automatic bifurcation diagram computation of stationary states. I will showcase this with the 2d Bratu problem.

Additionally, by leveraging on the above methods, the package can also seek for periodic orbits of Cauchy problems by casting them into an equation of high dimension. It is by now, one of the only software which provides parallel (Standard / Poincaré) shooting methods and finite differences based methods to compute periodic orbits in high dimensions. I will present an application highlighting the ability to fine tune BifurcationKit to get performance.

In a last part, I will describe a mean field model of stochastic spiking neurons described with a 2d measure valued equation. I will present a numerical scheme based on an implicit Finite Volume method. I will then provide some mathematical properties of the mean field concerning well posedness and stationary solutions. Additionally, I will show how BifurcationKit.jl can be used to study numerically the model. Finally, I will conclude on open problems, some of which could hopefully be tackled numerically with Trixi.jl.