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ExtGram.jl

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A discontinuous Galerkin implementation of the extended Gramian closure for moment equations with Trixi.jl.

Abstract (from the thesis)

The classical continuum mechanical equations of Navier-Stokes and Fourier (NSF) accurately describe gases with a high collision frequency, measured by a small Knudsen number (Kn). However, the NSF equations fail for gases with a high Knudsen number and, consequently, a small collision frequency. Directly solving Boltzmann's equation is an alternative approach of high accuracy, but is computationally very demanding due to its high dimensionality. Another method is to approximate the Boltzmann equation by directly solving for macroscopic properties, such as density, velocity, and temperature, which can be described by moments. Therefore, the so-called moment system is solved, which is an infinitely large system of hierarchically coupled partial differential equations (PDEs), with the moments as unknowns. The i-th PDE depends on the i-th moment and is coupled to the (i+1)-st PDE, again depending on the (i+1)-st moment. Directly solving the moment system is impossible, as it is infinitely large and always has one unknown more than equations. Therefore, the moment system is truncated after M equations, requiring a moment closure for the (M+1)-st moment.

This thesis discusses the extended Gramian closure, a recently invented moment closure based on orthogonal polynomials. The extended Gramian closure has attractive structure-preserving properties and overcomes shortcomings of classical closures, such as the lack of global hyperbolicity in Grad's closure and the computational effort required by the maximum entropy closure. This work introduces the theoretical framework of the extended Gramian closure and suggests an updated closure under the condition that the equations are truncated with M being odd. Rigorous mathematical proofs for the structure-preserving properties of this updated closure are discussed. While the closure is constructed in one dimension, a projection method for applying it to a three-dimensional velocity phase, based on matrix rotations, is introduced. Further, the extended Gramian closure is numerically solved with the discontinuous Galerkin method and its accuracy is analyzed and compared with that of the Gramian closure and Grad's closure. Therefore, various benchmark problems, including the shock tube, shock structure, Landau damping, and two-stream instability, are analyzed.

Main Features

  • Solving the 1D(space) - 1D(velocity) moment equations
    • Shock tube problem
    • Shock structure problem
    • Two beam problem was tested, but not yet included
  • Discrete velocity method (DVM) as reference solution for 1D(space) - 1D(velocity)
    • Shock tube
    • Shock structure
  • Extending the equations to 1D(space) - 3D(velocity) for M=4
    • Tested for the shock tube problem
  • Solving the 1D(space) - 1D(velocity) moment equations, coupled to the Vlasov-Poisson equation with initial conditions for:
    • Landau Damping
    • Two-stream instability
  • Numerical grid convergence
    • On periodic boundary conditions with initial conditions from Landau damping (no shock)

Installation

Clone the git repository and install the ExtGram.jl package locally with:

git clone git@git.rwth-aachen.de:JanHab/ExtGram.jl.git
cd ExtGram.jl
julia --project -e 'import Pkg; Pkg.instantiate()'

The solutions are stored in the out folder (in .gitignore). You need to create it and all subfolders. Further, Python (with Jupyter notebooks) is used for visualizations. Install the requirements from the requirements.txt file with (tested for Python version 3.12.2):

pip install -r requirements.txt
mkdir out
mkdir out/1D3D
mkdir out/1D3D/1D3D_angles
mkdir out/1D3D/Moments
mkdir out/CompareClosure
mkdir out/Figures
mkdir out/Figures/1D3D
mkdir out/Figures/ChiValues
mkdir out/Figures/CompareClosure
mkdir out/Figures/EigenvaluesEquilibrium
mkdir out/Figures/GaugeInvestigation
mkdir out/Figures/LandauDamping
mkdir out/Figures/ShockTube
mkdir out/Figures/ShockStructure
mkdir out/Figures/TwoStreamInstability
mkdir out/GridConvergencePeriodic
mkdir out/Shock
mkdir out/Shock/DVM
mkdir out/Shock/GaugeInvestigation
mkdir out/Shock/Moments
mkdir out/VlasovPoisson
mkdir out/VlasovPoisson/LandauDamping
mkdir out/VlasovPoisson/TwoStreamInstability

Testing

Clone the repository and execute the testing pipeline with:

julia --project test/tests.jl

Further, you can test the installation by running a shock tube example with:

julia --project examples/ShockTubeExample.jl

Usage

Find the package source code in src, which implements the momentum closure and the system of PDEs.

The folder examples contains the main files for physically relevant benchmark problems. slurm gives shell scripts for executing the benchmark problems either on your local laptop or on a HPC server with a slurm script. You need to either comment out the os.system(command) or the createsbatch file, depending on your system.

miscellaneous contains miscellaneous files used for implementation and checking.

notebooks has all the visualizations. The results are stored in out. Note that this folder is in .gitignore. The visualizations will only work if you have the necessary data. You can find minimal working data archived with Zenodo here: https://doi.org/10.5281/zenodo.18998361

Finally, test contains a small testing pipeline.

Referencing

@software{habscheid_2026_18998361,
  author       = {Habscheid, Jan and
                  Yilmaz, Eda and
                  Geratz, Matthias and
                  Torrilhon, Manuel},
  title        = {ExtGram.jl},
  month        = mar,
  year         = 2026,
  publisher    = {Zenodo},
  version      = {1.0.0},
  doi          = {10.5281/zenodo.18998361},
  url          = {https://doi.org/10.5281/zenodo.18998361},
}

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