Trixi.jl is a numerical simulation framework for 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.jl 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
  • Arbitrary floating-point precision
• Discontinuous Galerkin methods
  • Kinetic energy-preserving and entropy-stable methods based on flux differencing
  • Entropy-stable shock capturing
  • Finite difference summation by parts (SBP) methods
• Advanced limiting strategies
  • Positivity-preserving limiting
  • Subcell invariant domain-preserving (IDP) limiting
  • Entropy-bounded limiting
• 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
  • Compressible Navier-Stokes 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, LWR traffic flow)
• Multi-physics simulations
  • Self-gravitating gas dynamics
• Shared-memory parallelization via multithreading
• Multi-node parallelization via MPI
• 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

If you have not yet installed Julia, please follow the instructions for your operating system. Trixi.jl works with Julia v1.8 and newer. We recommend using the latest stable release of Julia.

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

julia> using 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.jl, while Plots.jl can be used to directly visualize Trixi.jl's results from the REPL.

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

julia> using 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.jl. A brief list of notable changes to Trixi.jl is available in NEWS.md.

If you plan on editing Trixi.jl itself, you can download Trixi.jl locally and use the code from the cloned directory:

git clone [email protected]:trixi-framework/Trixi.jl.git
cd Trixi.jl
mkdir run
cd run
julia --project=. -e 'using Pkg; Pkg.develop(PackageSpec(path=".."))' # Install local Trixi.jl clone
julia --project=. -e 'using Pkg; Pkg.add(["OrdinaryDiffEq", "Trixi2Vtk", "Plots"])' # Install additional packages

Note that the postprocessing tools Trixi2Vtk.jl and Plots.jl are optional and can be omitted.

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

julia --project=.

if already inside the run directory. Further details can be found in the documentation.

In the Julia REPL, first load the package Trixi.jl

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.jl elixir, i.e., a text file containing Julia code necessary to set up and run a simulation. To quickly see Trixi.jl 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.jl 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.

The presentation From Mesh Generation to Adaptive Simulation: A Journey in Julia, originally given as part of JuliaCon 2022, outlines how to use Trixi.jl 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.

Additional documentation is available that contains more information on how to modify/extend Trixi.jl'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.

If you use Trixi.jl in your own research or write a paper using results obtained with the help of Trixi.jl, 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.jl 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}
}

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

Trixi.jl is licensed under the MIT license (see LICENSE.md). Since Trixi.jl 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.

Participating research groups in alphabetical order:

Applied and Computational Mathematics, RWTH Aachen University 🇩🇪

Applied Mathematics, Department of Mathematics, University of Hamburg 🇩🇪

Division of Applied Mathematics, Department of Mathematics, Linköping University 🇸🇪

Computational and Applied Mathematics, Rice University 🇺🇸

High-Performance Computing, Institute of Software Technology, German Aerospace Center (DLR) 🇩🇪

High-Performance Scientific Computing, University of Augsburg 🇩🇪

Numerical Mathematics, Institute of Mathematics, Johannes Gutenberg University Mainz 🇩🇪

Numerical Simulation, Department of Mathematics and Computer Science, University of Cologne 🇩🇪

This project has benefited from funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the following grants:

• Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure.
• Research unit FOR 5409 "Structure-Preserving Numerical Methods for Bulk- and Interface Coupling of Heterogeneous Models (SNuBIC)" (project number 463312734).
• Individual grant no. 528753982.

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 through the VR Starting Grant "Shallow water flows including sediment transport and morphodynamics", VR grant agreement 2020-03642 VR.

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

This project has benefited from funding from the German Federal Ministry of Education and Research (BMBF) through the project grant "Adaptive earth system modeling with significantly reduced computation time for exascale supercomputers (ADAPTEX)" (funding id: 16ME0668K).

This project has benefited from funding by the Daimler und Benz Stiftung (Daimler and Benz Foundation) through grant no. 32-10/22.

Trixi.jl is supported by NumFOCUS as an Affiliated Project.