Trixi Framework

The Trixi framework is a collaborative scientific effort to provide open source tools for adaptive high-order numerical simulations of hyperbolic PDEs in Julia. Besides the core algorithms, the framework also includes mesh and visualization tools. Moreover, it includes utilities such as Julia wrappers of mature libraries written in other programming languages.

This page gives an overview of the different activities that, together, constitute the Trixi framework on GitHub.

Adaptive high-order numerical simulations of hyperbolic PDEs

• Trixi.jl

  Adaptive high-order numerical simulations of hyperbolic PDEs in Julia

• Trixi2Vtk.jl

  Convert output files generated with Trixi.jl to VTK

• libtrixi

  Use Trixi.jl from C/C++/Fortran

• SmartShockFinder.jl

  Create troubled cell indicators for Trixi.jl using artificial neural networks

Mesh generation

• HOHQMesh.jl

  HOHQMesh.jl is a Julia wrapper for the HOHQMesh mesh generator, which allows to produce curved quadrilateral and hexahedral meshes for high-order numerical simulations.

• HOHQMesh

  High Order Hex-Quad Mesh (HOHQMesh) package to automatically generate all-quadrilateral meshes with high order boundary information.

• Smesh.jl

  Smesh.jl is a Julia wrapper packagae for smesh, a simple Fortran package for generating and handling unstructured triangular and polygonal meshes.

• smesh

  A simple Fortran package for generating and handling unstructured triangular and polygonal meshes.

Particle-based multiphysics simulations

• TrixiParticles.jl

  Particle-based multiphysics simulations in Julia

• PointNeighbors.jl

  Efficient neighborhood search in point clouds with fixed search radius

Additional packages

• P4est.jl

  P4est.jl is lightweight Julia wrapper for the p4est C library.

• KROME.jl

  KROME.jl is a lightweight Julia wrapper for KROME, a Fortran library for including chemistry and microphysics in astrophysics simulations.

• JuliaVTK/ReadVTK.jl

  Julia package for reading VTK XML files (maintained by the Trixi framework authors).

Publications

The following publications make use of Trixi.jl or one of the other packages listed above. Author names of Trixi.jl's main developers are in italics.

2025

• Ranocha, Winters, Schlottke-Lakemper, Öffner, Glaubitz, Gassner, On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws, 2025.
  

2024

• Bach, Rueda-Ramírez, Kopriva, Gassner, Mimetic Metrics for the DGSEM, 2024.
  

• Doehring, Christmann, Schlottke-Lakemper, Gassner, Torrilhon, Fourth-Order Paired-Explicit Runge-Kutta Methods, 2024.
  

• Ersing, Goldberg, Winters, Entropy stable hydrostatic reconstruction schemes for shallow water systems, 2024.
  

• Glaubitz, Ranocha, Winters, Schlottke-Lakemper, Öffner, Gassner, Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods, 2024.
  

• Bender, Öffner, Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty, 2024.
  

• Oblapenko, Torrilhon, Entropy-conservative high-order methods for high-enthalpy flows, 2024.
  

• Doehring, Schlottke-Lakemper, Gassner, Torrilhon, Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics, 2024.
  

• Babbar, Chandrashekar, Multi-Derivative Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws, 2024.
  

• Lampert, Ranocha, Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations, 2024.
  

• Rueda-Ramírez, Sikstel, Gassner, An Entropy-Stable Discontinuous Galerkin Discretization of the Ideal Multi-Ion Magnetohydrodynamics System, 2024.
  

• Doehring, Gassner, Torrilhon, Many-Stage Optimal Stabilized Runge-Kutta Methods for Hyperbolic Partial Differential Equations, 2024.
  

• Babbar, Chandrashekar, Lax-Wendroff Flux Reconstruction for advection-diffusion equations with error-based time stepping, 2024.
  

• Babbar, Chandrashekar, Lax-Wendroff Flux Reconstruction on adaptive curvilinear meshes with error based time stepping for hyperbolic conservation laws, 2024.
  

2023

• Babbar, Kenettinkara, Chandrashekar, Admissibility preserving subcell limiter for Lax-Wendroff flux reconstruction, 2023.
  

• Ovadia, Oommen, Kahana, Peyvan, Turkel, Karniadakis, Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO), 2023.
  

• Ranocha, Schütz, Multiderivative time integration methods preserving nonlinear functionals via relaxation, 2023.
  

• Ranocha, Giesselmann, Stability of step size control based on a posteriori error estimates, 2023.
  

• Chan, Shukla, Wu, Liu, Nalluri, High order entropy stable schemes for the quasi-one-dimensional shallow water and compressible Euler equations, 2023.
  

• Ersing, Winters, An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes, 2023.
  

• Rueda-Ramírez, Bolm, Kuzmin, Gassner, Monolithic Convex Limiting for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods, 2023.
  

• Ranocha, A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators, 2023.
  

• Ranocha, Winters, Castro, Dalcin, Schlottke-Lakemper, Gassner, Parsani, On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics, 2023.
  

• Ranocha, Schlottke-Lakemper, Chan, Rueda-Ramírez, Winters, Hindenlang, Gassner, Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws, ACM Transactions on Mathematical Software, 2023.
  

2022

• Chan, Ranocha, Rueda-Ramírez, Gassner, Warburton, On the entropy projection and the robustness of high order entropy stable discontinuous Galerkin schemes for under-resolved flows, 2022.
  

• Rueda-Ramírez, Pazner, Gassner, Subcell limiting strategies for discontinuous Galerkin spectral element methods, 2022.
  

• Lukáčová-Medvid’ová, Öffner, Convergence of Discontinuous Galerkin Schemes for the Euler Equations via Dissipative Weak Solutions, 2022.
  

• Ranocha, A Note on Numerical Fluxes Conserving Harten's Entropies for the Compressible Euler Equations, 2022.
  

• Ranocha, Schlottke-Lakemper, Winters, Faulhaber, Chan, Gassner, Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing, JuliaCon Proceedings, 77, 2022.
  

• Gassner, Svärd, Hindenlang, Stability Issues of Entropy-Stable and/or Split-form High-order Schemes, 2022.
  

2021

• Singh, Chandrashekar, On a linear stability issue of split form schemes for compressible flows, 2021.
  

• Ranocha, Gassner, Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes, Communications on Applied Mathematics and Computation, 2021.
  

• Schlottke-Lakemper, Winters, Ranocha, Gassner, A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics, Journal of Computational Physics (442), 110467, 2021.
  

Talks

2024

• Non-intrusive Multirate Time-Integration for High-Order accurate Compressible Fluid Dynamics with Trixi.jl
  Doehring
  2nd July 2024, PDESoft 2024, Cambridge, UK

2023

• Challenges of sustainable research software engineering in Trixi.jl
  Schlottke-Lakemper
  27th October 2023, MBD Colloquium, Aachen, Germany

• Julia for scientific high-performance computing: opportunities and challenges
  Schlottke-Lakemper
  6th October 2023, Ferrite.jl User & Developer Conference, Bochum, Germany

• Scaling Trixi.jl to more than 10,000 cores using MPI
  Schlottke-Lakemper, Ranocha
  27th July 2023, JuliaCon 2023, Cambridge, US

• Massively Parallel Computational Fluid Dynamics with Julia and Trixi.jl
  Schlottke-Lakemper
  28th June 2023, PASC Conference, Davos, Switzerland

• Research Software Engineering for Sustainable Scientific Computing
  Schlottke-Lakemper
  30th January 2023, SSD Seminar Series, Aachen, Germany

• Trixi.jl: High-Order Numerical Simulations of Conservation Laws in Julia
  Schlottke-Lakemper
  19th January 2023, SNuBIC Seminar
  tutorials & notebooks

2022

• Robust and efficient high-performance computational fluid dynamics enabled by modern numerical methods and technologies
  Ranocha
  3rd November 2022, MUSEN Colloquium, TU Braunschweig, Germany

• Reproducibility as a service: collaborative scientific computing with Julia
  Schlottke-Lakemper, Ranocha
  27th October 2022, MaRDI Workshop for Scientific Computing, Münster, Germany

• From Mesh Generation to Adaptive Simulation: A Journey in Julia
  Winters
  27th July 2022, JuliaCon 2022
  recorded talk on YouTube | presentation & code

• Running Julia code in parallel with MPI: Lessons learned
  Christmann, Neher, Schlottke-Lakemper
  26th July 2022, Julia for HPC Minisymposium, JuliaCon 2022
  recorded talk on YouTube | presentation

• Extensible Computational Fluid Dynamics in Julia with Trixi.jl
  Schlottke-Lakemper, Ranocha, Gassner
  25th February 2022, SIAM Conference on Parallel Processing for Scientific Computing, Seattle, US

2021

• Research software development with Julia
  Schlottke-Lakemper, Ranocha
  27th September 2021, NFDI4Ing Conference 2021

• Adaptive high-order numerical simulations with Trixi.jl
  Schlottke-Lakemper, Ranocha
  9th September 2021, CliMA Seminar, California Institute of Technology

• Adaptive and extendable numerical simulations with Trixi.jl
  Schlottke-Lakemper, Ranocha
  30th July 2021, JuliaCon 2021
  presentation & notebooks | recorded talk on YouTube

• Trixi.jl: High-Order Numerical Simulations of Hyperbolic PDEs in Julia
  Ranocha, Schlottke-Lakemper, Winters
  14th July 2021, ICOSAHOM 2021
  tutorials & notebooks

• Introduction to Julia and Trixi, a numerical simulation framework for hyperbolic PDEs
  Ranocha
  27th April 2021, Applied Mathematics Seminar, University of Münster
  presentation

• Purely hyperbolic self-gravitating flow simulations in Julia
  Schlottke-Lakemper, Winters, Ranocha, Gassner
  15th March 2021, GAMM Annual Meeting 2021

• Julia for adaptive high-order multi-physics simulations
  Schlottke-Lakemper
  27th January 2021, Numerical Analysis Seminar, Lund University
  presentation & notebooks

Outreach

Google Summer of Code 2023

Trixi.jl participated in the Google Summer of Code 2023, marking its initial steps towards running on GPUs. This project was mentored by Hendrik Ranocha and Michael Schlottke-Lakemper. Here you can find the report from our contributor Huiyu Xie.

Authors

Michael Schlottke-Lakemper (University of Augsburg, Germany), Gregor Gassner (University of Cologne, Germany), Hendrik Ranocha (University of Hamburg, Germany), Andrew Winters (Linköping University, Sweden), and Jesse Chan (Rice University, US) are the principal developers of Trixi.jl. David A. Kopriva (Florida State University, US) is the principal developer of HOHQMesh and HOHQMesh.jl. For a full list of authors, please check out the respective packages.

Get in touch!

There are a number of ways to reach out to us:

• Meet us on Slack

• Create an issue in one of the repositories listed on this page

• Get in touch with one of the Trixi Authors

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.