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ryujin

About

ryujin is a high-performance, high-order collocation-type finite element solver for hyperbolic conservation equations, in particular the compressible Euler and Navier-Stokes equations of gas dynamics. The discretization builds on the invariant-domain preserving framework of Guermond and Popov: a first-order graph-viscosity update that keeps the numerical solution in the set of admissible states of the system, and the convex limiting technique that recovers second-order accuracy from it while enforcing the same local bounds. As a consequence the solver maintains the physical invariants of the underlying equations and is stable without ad-hoc tuning parameters.

The code is written in C++ on top of the deal.II finite element library. It runs on distributed-memory clusters via MPI, uses thread parallelism and explicit SIMD vectorization on the host, and executes its compute kernels on GPUs through deal.II and Kokkos.

License

Section titled “License”

ryujin is released under the terms of the Apache License 2.0 with LLVM Exception. Part of the contributed source code, third-party dependencies, and header libraries are covered by different open source licenses; see COPYING.md for details. Contributions are governed by the Developer Certificate of Origin 1.1, as described in CONTRIBUTING.md.

References

Section titled “References”

If you use ryujin for an academic publication, please cite some of the following references. The first group describes the numerical method and its implementation in ryujin; the second lists further work computed with the code; the third covers the mathematical framework the solver is based on.

Method and implementation

  • Structure-preserving finite-element approximations of the magnetic Euler-Poisson equationsJordan Hoffart, Matthias Maier, John Shadid, Ignacio TomasJournal of Computational Physics 566 (2026), 115269
    arXiv:2510.11808doi:10.1016/j.jcp.2026.115269
  • A conservative invariant-domain preserving projection technique for hyperbolic systems under adaptive mesh refinementJake Harmon, Martin Kronbichler, Matthias Maier, Eric J. TovarSIAM Journal on Scientific Computing, accepted 2026
    arXiv:2507.18717
  • Graph-based methods for hyperbolic systems of conservation laws using discontinuous space discretizationsMartin Kronbichler, Matthias Maier, Ignacio TomasCommunications in Computational Physics 38 (2025), 74–108
    arXiv:2402.04514doi:10.4208/cicp.OA-2024-0225
  • A high-order explicit Runge-Kutta approximation technique for the shallow water equationsJean-Luc Guermond, Matthias Maier, Eric J. TovarComputers & Fluids 288 (2025), 106493
    arXiv:2403.17123doi:10.1016/j.compfluid.2024.106493
  • First-order greedy invariant-domain preserving approximation for hyperbolic problems: scalar conservation laws, and p-systemJean-Luc Guermond, Matthias Maier, Bojan Popov, Laura Saavedra, Ignacio TomasJournal of Scientific Computing 100 (2024), 46
    arXiv:2310.01713doi:10.1007/s10915-024-02592-4
  • Robust second-order approximation of the compressible Euler equations with an arbitrary equation of stateBennett Clayton, Jean-Luc Guermond, Matthias Maier, Bojan Popov, Eric J. TovarJournal of Computational Physics 478 (2023), 111926
    arXiv:2207.12832doi:10.1016/j.jcp.2023.111926
  • On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier-Stokes equationsJean-Luc Guermond, Martin Kronbichler, Matthias Maier, Bojan Popov, Ignacio TomasComputer Methods in Applied Mechanics and Engineering 389 (2022), 114250
    arXiv:2106.02159doi:10.1016/j.cma.2021.114250
  • Efficient parallel 3D computation of the compressible Euler equations with an invariant-domain preserving second-order finite-element schemeMatthias Maier, Martin KronbichlerACM Transactions on Parallel Computing 8(3) (2021), 16:1–30
    arXiv:2007.00094doi:10.1145/3470637
  • Second-order invariant domain preserving approximation of the compressible Navier-Stokes equationsJean-Luc Guermond, Matthias Maier, Bojan Popov, Ignacio TomasComputer Methods in Applied Mechanics and Engineering 375 (2021), 113608
    arXiv:2009.06022doi:10.1016/j.cma.2020.113608

Further publications using ryujin

  • Guaranteed wave-speed bounds for the compressible Euler equations with a composite equation of stateNicolas Favrie, Matthias MaierSubmitted, 2026
    arXiv:2609.14287
  • Well-balanced second-order approximation of the compressible atmospheric Euler equationsCrystal Farris, Matthias Maier, Eric J. TovarSubmitted, 2026
    arXiv:2606.19764
  • Invariant-domain preserving IMEX schemes for the nonequilibrium gray radiation-hydrodynamics equations, Part IJean-Luc Guermond, Eric J. TovarSubmitted, 2026
    arXiv:2602.08291
  • Preserving the minimum principle on the entropy for the compressible Euler equations with general equations of stateBennett Clayton, Eric J. TovarJournal of Scientific Computing 108(3) (2026), 70
    arXiv:2503.10612doi:10.1007/s10915-026-03362-0
  • Second-order invariant-domain preserving approximation to the multi-species Euler equationsBennett Clayton, Tarik Dzanic, Eric J. TovarComputers & Fluids 317 (2026), 107159
    arXiv:2505.09581doi:10.1016/j.compfluid.2026.107159

Mathematical foundations

  • Second-order invariant domain preserving approximation of the Euler equations using convex limitingJean-Luc Guermond, Murtazo Nazarov, Bojan Popov, Ignacio TomasSIAM Journal on Scientific Computing 40(5) (2018), A3211–A3239
    arXiv:1710.00417doi:10.1137/17M1149961
  • Invariant domains and first-order continuous finite element approximation for hyperbolic systemsJean-Luc Guermond, Bojan PopovSIAM Journal on Numerical Analysis 54(4) (2016), 2466–2489
    arXiv:1509.07461doi:10.1137/16M1074291

Funding

Section titled “Funding”

Development of ryujin has been supported by the following grants.