Trixi.jl

Trixi.jl is a numerical simulation framework for hyperbolic conservation laws written in Julia. A key objective for the framework is to be useful to both scientists and students. Therefore, next to having an extensible design with a fast implementation, Trixi is focused on being easy to use for new or inexperienced users, including the installation and postprocessing procedures. Its features include:

• 1D, 2D, and 3D simulations on line/quad/hex/simplex meshes
  • Cartesian and curvilinear meshes
  • Conforming and non-conforming meshes
  • Structured and unstructured meshes
  • Hierarchical quadtree/octree grid with adaptive mesh refinement
  • Forests of quadtrees/octrees with p4est via P4est.jl
• High-order accuracy in space and time
• Discontinuous Galerkin methods
  • Kinetic energy-preserving and entropy-stable methods based on flux differencing
  • Entropy-stable shock capturing
  • Positivity-preserving limiting
  • Finite difference summation by parts (SBP) methods
• Compatible with the SciML ecosystem for ordinary differential equations
  • Explicit low-storage Runge-Kutta time integration
  • Strong stability preserving methods
  • CFL-based and error-based time step control
• Native support for differentiable programming
  • Forward mode automatic differentiation via ForwardDiff.jl
• Periodic and weakly-enforced boundary conditions
• Multiple governing equations:
  • Compressible Euler equations
  • Magnetohydrodynamics (MHD) equations
  • Multi-component compressible Euler and MHD equations
  • Linearized Euler and acoustic perturbation equations
  • Hyperbolic diffusion equations for elliptic problems
  • Lattice-Boltzmann equations (D2Q9 and D3Q27 schemes)
  • Shallow water equations
  • Several scalar conservation laws (e.g., linear advection, Burgers' equation)
• Multi-physics simulations
  • Self-gravitating gas dynamics
• Shared-memory parallelization via multithreading
• Visualization and postprocessing of the results
  • In-situ and a posteriori visualization with Plots.jl
  • Interactive visualization with Makie.jl
  • Postprocessing with ParaView/VisIt via Trixi2Vtk

Installation

If you have not yet installed Julia, please follow the instructions for your operating system. Trixi works with Julia v1.7.

For users

Trixi and its related tools are registered Julia packages. Hence, you can install Trixi, the visualization tool Trixi2Vtk, OrdinaryDiffEq.jl, and Plots.jl by executing the following commands in the Julia REPL:

julia> import Pkg

julia> Pkg.add(["Trixi", "Trixi2Vtk", "OrdinaryDiffEq", "Plots"])

You can copy and paste all commands to the REPL including the leading julia> prompts - they will automatically be stripped away by Julia. The package OrdinaryDiffEq.jl provides time integration schemes used by Trixi, while Plots.jl can be used to directly visualize Trixi's results from the REPL.

Note on package versions: If some of the examples for how to use Trixi do not work, verify that you are using a recent Trixi release by comparing the installed Trixi version from

julia> import Pkg; Pkg.update("Trixi"); Pkg.status("Trixi")

to the latest release. If the installed version does not match the current release, please check the Troubleshooting section in the documentation.

The commands above can also be used to update Trixi. A brief list of notable changes to Trixi is available in NEWS.md.

For developers

If you plan on editing Trixi itself, you can download Trixi locally and run it from within the cloned directory:

git clone git@github.com:trixi-framework/Trixi.jl.git
cd Trixi.jl
julia --project=@. -e 'import Pkg; Pkg.instantiate()' # Install Trixi's dependencies
julia -e 'import Pkg; Pkg.add(["Trixi2Vtk", "Plots"])' # Install postprocessing tools
julia -e 'import Pkg; Pkg.add("OrdinaryDiffEq")' # Install time integration schemes

If you installed Trixi this way, you always have to start Julia with the --project flag set to your local Trixi clone, e.g.,

julia --project=@.

Further details can be found in the documentation.

Usage

In the Julia REPL, first load the package Trixi

julia> using Trixi

Then start a simulation by executing

julia> trixi_include(default_example())

Please be patient since Julia will compile the code just before running it. To visualize the results, load the package Plots

julia> using Plots

and generate a heatmap plot of the results with

julia> plot(sol) # No trailing semicolon, otherwise no plot is shown

This will open a new window with a 2D visualization of the final solution:

The method trixi_include(...) expects a single string argument with the path to a Trixi elixir, i.e., a text file containing Julia code necessary to set up and run a simulation. To quickly see Trixi in action, default_example() returns the path to an example elixir with a short, two-dimensional problem setup. A list of all example elixirs packaged with Trixi can be obtained by running get_examples(). Alternatively, you can also browse the examples/ subdirectory. If you want to modify one of the elixirs to set up your own simulation, download it to your machine, edit the configuration, and pass the file path to trixi_include(...).

Note on performance: Julia uses just-in-time compilation to transform its source code to native, optimized machine code at the time of execution and caches the compiled methods for further use. That means that the first execution of a Julia method is typically slow, with subsequent runs being much faster. For instance, in the example above the first execution of trixi_include takes about 20 seconds, while subsequent runs require less than 60 milliseconds.

Showcase of advanced features

The presentation From Mesh Generation to Adaptive Simulation: A Journey in Julia, originally given as part of JuliaCon 2022, outlines how to use Trixi for an adaptive simulation of the compressible Euler equations in two spatial dimensions on a complex domain. More details as well as code to run the simulation presented can be found at the reproducibility repository for the presentation.

Documentation

Additional documentation is available that contains more information on how to modify/extend Trixi's implementation, how to visualize output files etc. It also includes a section on our preferred development workflow and some tips for using Git. The latest documentation can be accessed either online or under docs/src.

Referencing

If you use Trixi in your own research or write a paper using results obtained with the help of Trixi, please cite the following articles:

@article{ranocha2022adaptive,
  title={Adaptive numerical simulations with {T}rixi.jl:
         {A} case study of {J}ulia for scientific computing},
  author={Ranocha, Hendrik and Schlottke-Lakemper, Michael and Winters, Andrew Ross
          and Faulhaber, Erik and Chan, Jesse and Gassner, Gregor},
  journal={Proceedings of the JuliaCon Conferences},
  volume={1},
  number={1},
  pages={77},
  year={2022},
  doi={10.21105/jcon.00077},
  eprint={2108.06476},
  eprinttype={arXiv},
  eprintclass={cs.MS}
}

@article{schlottkelakemper2021purely,
  title={A purely hyperbolic discontinuous {G}alerkin approach for
         self-gravitating gas dynamics},
  author={Schlottke-Lakemper, Michael and Winters, Andrew R and
          Ranocha, Hendrik and Gassner, Gregor J},
  journal={Journal of Computational Physics},
  pages={110467},
  year={2021},
  month={06},
  volume={442},
  publisher={Elsevier},
  doi={10.1016/j.jcp.2021.110467},
  eprint={2008.10593},
  eprinttype={arXiv},
  eprintclass={math.NA}
}

In addition, you can also refer to Trixi directly as

@misc{schlottkelakemper2020trixi,
  title={{T}rixi.jl: {A}daptive high-order numerical simulations
         of hyperbolic {PDE}s in {J}ulia},
  author={Schlottke-Lakemper, Michael and Gassner, Gregor J and
          Ranocha, Hendrik and Winters, Andrew R and Chan, Jesse},
  year={2021},
  month={09},
  howpublished={\url{https://github.com/trixi-framework/Trixi.jl}},
  doi={10.5281/zenodo.3996439}
}

Authors

Trixi was initiated by Michael Schlottke-Lakemper (RWTH Aachen University, Germany) and Gregor Gassner (University of Cologne, Germany). Together with Hendrik Ranocha (University of Hamburg, Germany), Andrew Winters (Linköping University, Sweden), and Jesse Chan (Rice University, US), they are the principal developers of Trixi. The full list of contributors can be found in AUTHORS.md.

License and contributing

Trixi is licensed under the MIT license (see LICENSE.md). Since Trixi is an open-source project, we are very happy to accept contributions from the community. Please refer to CONTRIBUTING.md for more details. Note that we strive to be a friendly, inclusive open-source community and ask all members of our community to adhere to our CODE_OF_CONDUCT.md. To get in touch with the developers, join us on Slack or create an issue.

Acknowledgments

This project has benefited from funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure.

This project has benefited from funding from the European Research Council through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws" (Extreme), ERC grant agreement no. 714487.

This project has benefited from funding from Vetenskapsrådet (VR, Swedish Research Council), Sweden grant agreement 2020-03642 VR.

This project has benefited from funding from the United States National Science Foundation under awards DMS-1719818 and DMS-1943186.