# Trixi.jl

**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
- 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
- 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
- Scalar advection

- Multi-physics simulations
- Shared-memory parallelization via multithreading
- Multi-node parallelization via MPI
- Visualization and postprocessing of the results

## Installation

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.

### For users

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.

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`

.

### For developers

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

```
git clone git@github.com:trixi-framework/Trixi.jl.git
cd Trixi.jl
mkdir run
cd run
julia --project=. -e 'using Pkg; Pkg.develop(PackageSpec(path=".."))'
```

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.

The advantage of using a separate `run`

directory is that you can also add other related packages (see below, e.g., for time integration or visualization) to the project in the `run`

folder and always have a reproducible environment at hand to share with others.

Since the postprocessing tool Trixi2Vtk.jl typically does not need to be modified, it is recommended to install it as a normal package. Likewise, you can install OrdinaryDiffEq.jl and Plots.jl as ordinary packages. To achieve this, use the following REPL commands:

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

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

### Example: Installing Trixi.jl as a package

Please note that the playback speed is set to 3x, thus the entire installation procedure lasts around 45 seconds in real time (depending on the performance of your computer and on how many dependencies had already been installed before).

## Usage

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(...)`

.

### Example: Running a simulation with Trixi.jl

If this produces weird symbols or question marks in the terminal on your system, you are probably using Mac OS with problematic fonts. In that case, please check the Troubleshooting section.

*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 *milli*seconds.

### Performing a convergence analysis

To automatically determine the experimental order of convergence (EOC) for a given setup, execute

`julia> convergence_test(default_example(), 4)`

This will run a convergence test with the elixir `default_example()`

, using four iterations with different initial refinement levels. The initial iteration will use the elixir unchanged, while for each subsequent iteration the `initial_refinement_level`

parameter is incremented by one. Finally, the measured $l^2$ and $l^\infty$ errors and the determined EOCs will be displayed like this:

```
[...]
l2
scalar
error EOC
9.14e-06 -
5.69e-07 4.01
3.55e-08 4.00
2.22e-09 4.00
mean 4.00
--------------------------------------------------------------------------------
linf
scalar
error EOC
6.44e-05 -
4.11e-06 3.97
2.58e-07 3.99
1.62e-08 4.00
mean 3.99
--------------------------------------------------------------------------------
```

An example with multiple variables looks like this:

`julia> convergence_test(joinpath(examples_dir(), "tree_2d_dgsem", "elixir_euler_source_terms.jl"), 3)`

```
[...]
l2
rho rho_v1 rho_v2 rho_e
error EOC error EOC error EOC error EOC
9.32e-07 - 1.42e-06 - 1.42e-06 - 4.82e-06 -
7.03e-08 3.73 9.53e-08 3.90 9.53e-08 3.90 3.30e-07 3.87
4.65e-09 3.92 6.09e-09 3.97 6.09e-09 3.97 2.12e-08 3.96
mean 3.82 mean 3.93 mean 3.93 mean 3.91
--------------------------------------------------------------------------------
linf
rho rho_v1 rho_v2 rho_e
error EOC error EOC error EOC error EOC
9.58e-06 - 1.17e-05 - 1.17e-05 - 4.89e-05 -
6.23e-07 3.94 7.48e-07 3.97 7.48e-07 3.97 3.22e-06 3.92
4.05e-08 3.94 4.97e-08 3.91 4.97e-08 3.91 2.10e-07 3.94
mean 3.94 mean 3.94 mean 3.94 mean 3.93
--------------------------------------------------------------------------------
```

### 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.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.

## Referencing

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}
}
```

## Authors

Trixi.jl was initiated by Michael Schlottke-Lakemper (RWTH Aachen University/High-Performance Computing Center Stuttgart (HLRS), Germany) and Gregor Gassner (University of Cologne, Germany). Together with Hendrik Ranocha (Johannes Gutenberg University Mainz, Germany) and 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 under Authors.

## License and contributing

Trixi.jl is licensed under the MIT license (see License). Since Trixi.jl is an open-source project, we are very happy to accept contributions from the community. Please refer to Contributing 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. 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) 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.