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arxiv: 2605.28627 · v1 · pith:R54UQYSXnew · submitted 2026-05-27 · ⚛️ physics.flu-dyn

An Architecture-Agnostic High-Order Discontinuous Galerkin Framework for Compressible Flows

Spencer Starr , Yannik Feldner , Patrick Kopper , Marcel Blind , Daniel Kempf , Jannik Schrempp , Felix Rodach , Andrea Beck
show 1 more author
Anna Schwarz
This is my paper

Pith reviewed 2026-06-29 09:49 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords discontinuous Galerkin spectral element methodGPU accelerationcompressible flowsarchitecture-agnostichigh-order methodslarge eddy simulationparallel scalingTaylor-Green vortex
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The pith

GALÆXI delivers an architecture-agnostic high-order DGSEM framework for compressible flows that runs on both NVIDIA and AMD GPUs from a single Fortran codebase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces GALÆXI as a high-order discontinuous Galerkin spectral element method framework for studying complex compressible turbulent flows on unstructured hexahedral grids. GPU acceleration is achieved by interfacing the Fortran source to CUDA C++ for NVIDIA hardware and HIP C++ for AMD hardware, enabling a single codebase across vendors. The implementation is verified for expected convergence order using manufactured solutions and matches reference results for the compressible Taylor-Green vortex across architectures. It demonstrates near-ideal scaling to 65,536 AMD GPUs for a 67.1 billion degree-of-freedom simulation at 82.6% efficiency, with GPU runs providing 7.75x to 8.08x speedups over CPU while using less than half the energy. Production-scale wall-resolved large eddy simulations of transonic airfoil flows with shock buffet conditions further illustrate its capability.

Core claim

GALÆXI is an open source architecture-agnostic toolchain for high-order DGSEM computations of complex compressible turbulent flows on unstructured hexahedral grids. GPU-accelerated computations are enabled by interfacing Fortran source code to CUDA C++ for NVIDIA and HIP C++ for AMD. The DGSEM implementation is verified to the expected order of convergence via the method of manufactured solutions and shows excellent agreement with reference solutions for the compressible Taylor-Green vortex across all supported architectures. Near-ideal strong and weak scaling is achieved on both NVIDIA and AMD GPU hardware, including a simulation with 67.1 billion degrees of freedom on 65,536 AMD MI250X dev

What carries the argument

The Fortran-to-CUDA and Fortran-to-HIP interfaces that allow a single DGSEM source code to execute on multiple GPU architectures while preserving the high-order discretization on unstructured hexahedral grids.

If this is right

  • High-order accuracy and agreement with reference solutions hold across both NVIDIA and AMD GPU hardware for verification cases.
  • Near-ideal strong and weak scaling is maintained up to at least 65,536 AMD MI250X devices for simulations exceeding 67 billion degrees of freedom.
  • GPU-based runs deliver 7.75x to 8.08x speedups over CPU computations in time-to-solution while using less than half the energy per node.
  • Wall-resolved large eddy simulations of transonic flows past airfoils with shock buffet are feasible at production scale on the supported architectures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar interface layers could allow other high-order CFD codes to support additional GPU vendors without full rewrites.
  • Users could run identical simulations on whichever hardware is available or cheaper at a given facility.
  • The framework opens the possibility of direct numerical comparisons between results computed on different GPU architectures to check for implementation-specific biases.

Load-bearing premise

The Fortran-to-CUDA and Fortran-to-HIP interfaces maintain numerical correctness and performance portability without hidden vendor-specific optimizations that would break the architecture-agnostic claim.

What would settle it

A side-by-side run of the method of manufactured solutions on NVIDIA and AMD hardware that produces different observed orders of convergence or solution errors for the same problem setup.

Figures

Figures reproduced from arXiv: 2605.28627 by Andrea Beck, Anna Schwarz, Daniel Kempf, Felix Rodach, Jannik Schrempp, Marcel Blind, Patrick Kopper, Spencer Starr, Yannik Feldner.

Figure 1
Figure 1. Figure 1: Evaluation timeline of the entropy-stable DGSEM from Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the integral kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Center plane slice of a Schlieren image of the flow field for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Strong (left) and weak (right) scaling performance of GALÆXI on GPU-accelerated HPC systems. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Strong (left) and weak (right) scaling performance of GALÆXI on CPU-based HPC systems. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Isosurface of the instantaneous Mach number at [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the results for the CPU and AMD GPU [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of the flow around ONERA OAT15A airfoil at [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

With the recent proliferation of heterogeneous, GPU-accelerated supercomputers, high-order computational fluid dynamics (CFD) simulations of complex, turbulent flows are more accessible than ever. To leverage the computing power of these machines, CFD software must adapt. However, complicating the situation is the emerging need to support hardware from multiple GPU vendors. Addressing this need is the GPU-accelerated, discontinuous Galerkin spectral element method (DGSEM) framework GAL{\AE}XI, a high-order, open source, architecture-agnostic toolchain for the study of complex, compressible, turbulent flows on unstructured, hexahedral grids. GPU-accelerated computations with GAL{\AE}XI are possible on GPU hardware by interfacing Fortran source code to the vendor models CUDA C++ for NVIDIA and HIP C++ for AMD. The DGSEM implementation in GAL{\AE}XI was verified using the method of manufactured solutions to rigorously confirm the expected order of convergence. Simulations of a compressible Taylor-Green-Vortex also demonstrated excellent agreement with reference solutions across all supported architectures. GAL{\AE}XI achieved near ideal strong and weak scaling on GPU hardware from both NVIDIA and AMD. In the largest case, GAL{\AE}XI performed a simulation with 67.1 billion degrees of freedom on 65,536 AMD MI250X graphics compute devices with a parallel efficiency of 82.6%. Comparing node-to-node performance, GPU simulations offered speedups between 7.75x and 8.08x over CPU computations in time-to-solution while consuming less than half the energy. To demonstrate GAL{\AE}XI's effectiveness for production-scale simulations, wall-resolved large eddy simulations of the transonic flow past a NACA 64A-110 airfoil and an ONERA OAT15A airfoil under shock buffet conditions were computed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces GALÆXI, an open-source DGSEM framework for compressible flows on unstructured hexahedral grids that achieves architecture-agnostic GPU acceleration by interfacing Fortran source to CUDA C++ (NVIDIA) and HIP C++ (AMD). The DGSEM implementation is verified via the method of manufactured solutions for expected convergence order and tested on the compressible Taylor-Green vortex, showing agreement with reference solutions across architectures. The work reports near-ideal strong and weak scaling, including a 67.1-billion-DOF simulation on 65,536 AMD MI250X devices at 82.6% parallel efficiency, node-to-node GPU speedups of 7.75–8.08× over CPU with less than half the energy consumption, and demonstrates production use via wall-resolved LES of transonic airfoil flows under shock-buffet conditions.

Significance. If the results hold, the contribution is significant for enabling high-order CFD on multi-vendor heterogeneous supercomputers. The open-source release, MMS verification, cross-architecture TGV agreement, and the demonstrated 67.1B-DOF run with quantified efficiency and energy metrics provide concrete evidence of correctness and performance portability that directly supports the central architecture-agnostic claim.

minor comments (2)
  1. The TGV results are described as showing 'excellent agreement' with references; adding quantitative error norms or L2-error tables (e.g., in §4 or a dedicated results subsection) would allow readers to assess the precise level of match across architectures.
  2. The node-to-node speedup and energy comparisons (7.75–8.08×, <½ energy) would benefit from explicit specification of the CPU baseline hardware and compiler flags to ensure reproducibility of the performance claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance for multi-vendor GPU CFD, and recommendation to accept. No major comments were raised, so we have no points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is an implementation and benchmarking paper for the GALÆXI DGSEM framework. It verifies the method via the method of manufactured solutions (standard external check) and reports agreement with independent reference solutions for Taylor-Green-Vortex and airfoil cases. Scaling, performance, and energy results are direct measurements on hardware, not derived predictions. No equations, fitted parameters, or uniqueness claims reduce to self-inputs or self-citations by construction. The architecture-agnostic claim rests on cross-vendor numerical agreement rather than unanchored assertion.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software framework and performance paper; no free parameters, mathematical axioms, or invented physical entities are introduced. The central claims rest on standard numerical methods and hardware interfaces.

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Works this paper leans on

85 extracted references · 21 canonical work pages

  1. [1]

    Accessed: 2026-14-01

    TOP500 - November 2025, https://www.top500.org/lists/ top500/list/2025/11/, 2025. Accessed: 2026-14-01

    2025

  2. [2]

    Accessed: 2026-14-01

    TOP500 - November 2020, https://www.top500.org/lists/ top500/list/2020/11/, 2020. Accessed: 2026-14-01

    2020

  3. [3]

    Gasparino, F

    L. Gasparino, F. Spiga, O. Lehmkuhl, SOD2D: A GPU-enabled spectral finite elements method for compressible scale-resolving simulations, Computer Physics Communications 297 (2024) 109067

    2024

  4. [4]

    F. D. Witherden, P. E. Vincent, W. Trojak, Y. Abe, A. Ak- barzadeh, S. Akkurt, M. Alhawwary, L. Caros, T. Dzanic, G. Giangaspero, et al., PyFR v2.0.3: Towards industrial adop- tion of scale-resolving simulations, Computer Physics Commu- nications 311 (2025) 109567

    2025

  5. [5]

    Eleftherakis, G

    P.-E. Eleftherakis, G. Anagnostopoulos, A. Kapetanakis, M. Umair, J.-Y. Vet, K. Iliakis, J. Vincent, J. Gong, R. Vin- uesa, S. Xydis, POSTER: Performance Portability in GPU- Accelerated Spectral Finite Element Fluid Simulations: A Cross-layer Exploration Approach, in: Proceedings of the 22nd ACM International Conference on Computing Frontiers, 2025, pp. 228–229

    2025

  6. [6]

    Dzanic, F

    T. Dzanic, F. Witherden, Positivity-preserving entropy-based adaptive filtering for discontinuous spectral element methods, Journal of Computational Physics 468 (2022) 111501. doi: https: //doi.org/10.1016/j.jcp.2022.111501

    work page doi:10.1016/j.jcp.2022.111501 2022

  7. [7]

    makotemplates.org/, 2026

    Mako, Mako Templates for Python, https://www. makotemplates.org/, 2026. Accessed: 2026-23-04

    2026

  8. [8]

    Industrial LES

    K. Fujii, S. Kawai, D. Gaitonde, Scale Resolving Methods for Aeronautical Flows toward the Era of “Industrial LES”, Flow, Turbulence and Combustion 115 (2025) 405–446

    2025

  9. [9]

    A. Ceci, A. Palumbo, S. Pirozzoli, Grid resolution requirements for dns of shock/boundary-layer interactions, Computers & Flu- ids (2025) 106892

    2025

  10. [10]

    Zeifang, A

    J. Zeifang, A. Beck, A data-driven high order sub-cell artifi- cial viscosity for the discontinuous Galerkin spectral element method, Journal of Computational Physics 441 (2021) 110475. doi:10.1016/j.jcp.2021.110475

    work page doi:10.1016/j.jcp.2021.110475 2021

  11. [11]

    Herten, Many Cores, Many Models: GPU Programming Model vs

    A. Herten, Many Cores, Many Models: GPU Programming Model vs. Vendor Compatibility Overview, in: Proceedings of the SC ’23 Workshops of the International Conference on High Performance Computing, Network, Storage, and Analysis, SC- W ’23, Association for Computing Machinery, 2023, pp. 1019–

    2023

  12. [12]

    doi: 10.1145/3624062.3624178

    work page doi:10.1145/3624062.3624178

  13. [13]

    Fischer, J

    P. Fischer, J. Lottes, H. Tufo, Nek5000, Technical Report, Ar- gonne National Lab, Argonne, IL (United States), 2007

    2007

  14. [14]

    De Vanna, F

    F. De Vanna, F. A vanzi, M. Cogo, S. Sandrin, M. Bettencourt, F. Picano, E. Benini, URANOS: A GPU accelerated Navier- Stokes solver for compressible wall-bounded flows, Computer Physics Communications 287 (2023) 108717

    2023

  15. [15]

    Jansson, M

    N. Jansson, M. Karp, A. Podobas, S. Markidis, P. Schlatter, Neko: A modern, portable, and scalable framework for high- fidelity computational fluid dynamics, Computers & Fluids (2024) 106243

    2024

  16. [16]

    Krais, A

    N. Krais, A. Beck, T. Bolemann, H. Frank, D. Flad, G. Gassner, F. Hindenlang, M. Hoffmann, T. Kuhn, M. Sonntag, C.-D. Munz, FLEXI: A high order discontinuous Galerkin frame- work for hyperbolic–parabolic conservation laws, Computers & Mathematics with Applications 81 (2021) 186–219

    2021

  17. [17]

    Rubio, HORSES3D: a high-order discontinuous Galerkin solver for flow simulations and multi-physics applications, arXiv (Cornell University) (2022)

    G. Rubio, HORSES3D: a high-order discontinuous Galerkin solver for flow simulations and multi-physics applications, arXiv (Cornell University) (2022)

    2022

  18. [18]

    Fischer, S

    P. Fischer, S. Kerkemeier, M. Min, Y.-H. Lan, M. Phillips, T. Rathnayake, E. Merzari, A. Tomboulides, A. Karakus, N. Chalmers, et al., NekRS, a GPU-accelerated spectral element Navier–Stokes solver, Parallel Computing 114 (2022) 102982

    2022

  19. [19]

    Accessed: 2026-11-03

    NVIDIA, CUDA Toolkit, https://developer.nvidia.com/ cuda-toolkit, 2026. Accessed: 2026-11-03

    2026

  20. [20]

    Accessed: 2026-11-03

    Advanced Micro Devices, HIP, https://rocm.docs.amd.com/ projects/HIP/en/latest/, 2026. Accessed: 2026-11-03

    2026

  21. [21]

    Blind, M

    M. Blind, M. Gao, D. Kempf, P. Kopper, M. Kurz, A. Schwarz, A. Beck, Towards Exascale CFD Simulations Using the Discon- tinuous Galerkin Solver FLEXI, Springer Nature Switzerland, 2026, pp. 207–221. doi: 10.1007/978-3-031-91312-9_15

    work page doi:10.1007/978-3-031-91312-9_15 2026

  22. [22]

    M. Kurz, D. Kempf, M. Blind, P. Kopper, P. Offenhäuser, A. Schwarz, S. Starr, J. Keim, A. Beck, GALÆXI: Solving com- plex compressible flows with high-order discontinuous Galerkin methods on accelerator-based systems, Computer Physics Com- munications 306 (2024). doi: 10.1016/j.cpc.2024.109388

    work page doi:10.1016/j.cpc.2024.109388 2024

  23. [23]

    Sonntag, C.-D

    M. Sonntag, C.-D. Munz, Shock capturing for discontinuous Galerkin methods using finite volume subcells, in: Finite Vol- umes for Complex Applications VII-Elliptic, Parabolic and Hy- perbolic Problems: FVCA 7, Berlin, June 2014, Springer, 2014, pp. 945–953

    2014

  24. [24]

    A. D. Beck, T. Bolemann, D. Flad, H. Frank, G. J. Gassner, F. Hindenlang, C.-D. Munz, High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations, International Journal for Numerical Methods in Fluids 76 (2014) 522–548

    2014

  25. [25]

    J. Keim, A. Schwarz, P. Kopper, M. Blind, C. Rohde, A. Beck, Entropy Stable High-Order Discontinuous Galerkin Spectral- Element Methods on Curvilinear, Hybrid Meshes, Journal of Computational Physics 557 (2026) 114829. doi: 10.1016/j.jcp. 2026.114829

    work page doi:10.1016/j.jcp 2026

  26. [26]

    Mossier, P

    P. Mossier, P. Oestringer, S. Jöns, J. Keim, C. Mavriplis, A. D. Beck, C.-D. Munz, Tackling Compressible Turbulent Multi- Component Flows with Dynamic Hp-Adaptation 306 (2026) 106928. doi: 10.1016/j.compfluid.2025.106928

    work page doi:10.1016/j.compfluid.2025.106928 2026

  27. [27]

    Accessed: 2026-26- 02

    OpenMP, https://www.openmp.org/, 2026. Accessed: 2026-26- 02

    2026

  28. [28]

    Accessed: 2026- 26-02

    OpenACC, https://www.openacc.org/, 2026. Accessed: 2026- 26-02

    2026

  29. [29]

    C. R. Trott, D. Lebrun-Grandié, D. Arndt, J. Ciesko, V. Dang, N. Ellingwood, R. Gayatri, E. Harvey, D. S. Hollman, D. Ibanez, N. Liber, J. Madsen, J. Miles, D. Poliakoff, A. Powell, S. Raja- manickam, M. Simberg, D. Sunderland, B. Turcksin, J. Wilke, Kokkos 3: Programming Model Extensions for the Exascale Era, IEEE Transactions on Parallel and Distributed...

    work page doi:10.1109/tpds.2021.3097283 2022

  30. [30]

    Lawrence Livermore National Laboratory, RAJA Portabil- ity Suite: Enabling Performance Portable CPU and GPU HPC Applications, https://computing.llnl.gov/projects/ raja-managing-application-portability-next-generation\ protect\penalty\z@-platforms, 2026

    2026

  31. [31]

    Zenker, B

    E. Zenker, B. Worpitz, R. Widera, A. Huebl, G. Juckeland, A. Knüpfer, W. E. Nagel, M. Bussmann, Alpaka – An Ab- straction Library for Parallel Kernel Acceleration, in: 2016 IEEE International Parallel and Distributed Processing Sym- posium Workshops (IPDPSW), 2016, pp. 631–640. doi: 10.1109/ IPDPSW.2016.50

    2016

  32. [32]

    Accessed: 2026-11-03

    Khronos Group, SYCL, https://www.khronos.org/sycl/, 2026. Accessed: 2026-11-03

    2026

  33. [33]

    Accessed: 2026-26-02

    NVIDIA, CUDA Fortran, https://developer.nvidia.com/ cuda-fortran, 2026. Accessed: 2026-26-02

    2026

  34. [34]

    Accessed: 2026-15-04

    Fortran Community, Fortran: High-performance parallel pro- gramming language, https://fortran-lang.org/, 2026. Accessed: 2026-15-04

    2026

  35. [35]

    S. Bak, C. Bertoni, S. Boehm, R. Budiardja, B. M. Chap- man, J. Doerfert, M. Eisenbach, H. Finkel, O. Hernandez, J. Huber, S. Iwasaki, V. Kale, P. R. Kent, J. Kwack, M. Lin, P. Luszczek, Y. Luo, B. Pham, S. Pophale, K. Ravikumar, V. Sarkar, T. Scogland, S. Tian, P. Yeung, OpenMP application experiences: Porting to accelerated nodes, Parallel Computing 18 ...

    work page doi:10.1016/j.parco.2021.102856 2022

  36. [36]

    Bernardini, D

    M. Bernardini, D. Modesti, F. Salvadore, S. Pirozzoli, STREAmS: A high-fidelity accelerated solver for direct nu- merical simulation of compressible turbulent flows, Computer Physics Communications 263 (2021) 107906

    2021

  37. [37]

    Salvadore, G

    F. Salvadore, G. Rossi, S. Sathyanarayana, M. Bernardini, OpenMP offload toward the exascale using Intel® GPU Max 1550: evaluation of STREAmS compressible solver, The Jour- nal of Supercomputing 80 (2024) 21094–127

    2024

  38. [38]

    De Vanna, G

    F. De Vanna, G. Baldan, URANOS-2.0: Improved performance, enhanced portability, and model extension towards exascale computing of high-speed engineering flows, Computer Physics Communications 303 (2024) 109285

    2024

  39. [39]

    Solanki, AdaptiveCpp design and architecture, https://github.com/AdaptiveCpp/AdaptiveCpp/blob/ develop/doc/architecture.md, 2025

    M. Solanki, AdaptiveCpp design and architecture, https://github.com/AdaptiveCpp/AdaptiveCpp/blob/ develop/doc/architecture.md, 2025. Accessed: 2026-09-03

    2025

  40. [40]

    Z. Dai, L. Deng, Y. Che, M. Li, J. Zhang, Y. Wang, Evalu- ating performance portability of five shared-memory program- ming models using a high-order unstructured CFD solver, Jour- nal of Parallel and Distributed Computing 187 (2024) 104831. doi:j.jpdc.2023.104831

    work page arXiv 2024

  41. [41]

    Bauman, B

    P. Bauman, B. Messer, J. Glenski, A. Georgiadou, J. Li- etz, K. Gottiparthi, M. Day, J. Chen, J. Rood, L. Esclapez, J. White III, G. R. Jansen, N. Curtis, S. Nichols, J. Kurzak, N. Chalmers, C. Freitag, N. Malaya, A. Fanfarillo, R. D. Bu- diardja, T. Papatheodore, N. Frontiere, D. Mcdougall, M. Nor- man, S. Sreepathi, P. Roth, D. Bykov, N. Wolfe, P. Mullo...

    work page doi:10.1145/3581784.3607065 2023

  42. [42]

    Nastac, A

    G. Nastac, A. Walden, E. J. Nielsen, K. Frendi, Implicit Thermochemical Nonequilibrium Flow Simulations on Unstruc- tured Grids using GPUs, in: AIAA Scitech 2021 Forum, 2021. doi:10.2514/6.2021-0159

    work page doi:10.2514/6.2021-0159 2021

  43. [43]

    Jansson, M

    N. Jansson, M. Karp, J. Wahlgren, S. Markidis, P. Schlatter, Design of Neko—A Scalable High-Fidelity Simulation Frame- work With Extensive Accelerator Support, Concurrency and Computation: Practice and Experience 37 (2025)

    2025

  44. [44]

    W. K. Anderson, R. T. Biedron, J.-R. Carlson, J. M. Derlaga, B. Diskin, C. T. Druyor Jr., et al, FUN3D Manual: 14.2, 2025

    2025

  45. [45]

    Nastac, A

    G. Nastac, A. Walden, L. Wang, E. J. Nielsen, Y. Liu, M. Opgenorth, J. Orender, M. Zubair, A Multi-Architecture Approach for Implicit Computational Fluid Dynamics on Un- structured Grids, in: AIAA Scitech 2023 Forum, 2023. doi: 10. 2514/6.2023-1226

    2023

  46. [46]

    Harten, J

    A. Harten, J. M. Hyman, Self Adjusting Grid Methods for One- Dimensional Hyperbolic Conservation Laws, Journal of Compu- tational Physics 50 (1983) 235–269. doi: 10.1016/0021-9991(83) 90066-9

    work page doi:10.1016/0021-9991(83 1983

  47. [47]

    Bassi, S

    F. Bassi, S. Rebay, A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics 131 (1997) 267–279. doi: 10.1006/jcph.1996.5572

    work page doi:10.1006/jcph.1996.5572 1997

  48. [48]

    P. Chandrashekar, Kinetic Energy Preserving and Entropy Sta- ble Finite Volume Schemes for Compressible Euler and Navier- Stokes Equations, Communications in Computational Physics 14 (2013) 1252–1286. doi: 10.4208/cicp.170712.010313a

    work page doi:10.4208/cicp.170712.010313a 2013

  49. [49]

    Sonntag, Shape Derivatives and Shock Capturing for the Navier-Stokes Equations in Discontinuous Galerkin Meth- ods, Ph.D

    M. Sonntag, Shape Derivatives and Shock Capturing for the Navier-Stokes Equations in Discontinuous Galerkin Meth- ods, Ph.D. thesis, Universität Stuttgart, 2017. doi: 10.18419/ opus-9342

    2017

  50. [50]

    Persson, J

    P.-O. Persson, J. Peraire, Sub-Cell Shock Capturing for Discon- tinuous Galerkin Methods, 2006. doi: 10.2514/6.2006-112

    work page doi:10.2514/6.2006-112 2006

  51. [51]

    M. H. Carpenter, A. Kennedy, Fourth-Order 2N-Storage Runge- Kutta Schemes, Nasa Technical Memorandum 109112 (1994) 1–26

    1994

  52. [52]

    Niegemann, R

    J. Niegemann, R. Diehl, K. Busch, Efficient Low-Storage Runge–Kutta Schemes with Optimized Stability Regions, Jour- nal of Computational Physics 231 (2012) 364–372. doi: 10.1016/ j.jcp.2011.09.003

    2012

  53. [53]

    Schwarz, D

    A. Schwarz, D. Kempf, J. Keim, P. Kopper, C. Rohde, A. Beck, Comparison of Entropy Stable Collocation High-Order DG Methods for Compressible Turbulent Flows, Computers & Flu- ids 303 (2025) 106874. doi: 10.1016/j.compfluid.2025.106874

    work page doi:10.1016/j.compfluid.2025.106874 2025

  54. [54]

    Reid, The new features of Fortran 2003 26 (2007) 10–33

    J. Reid, The new features of Fortran 2003 26 (2007) 10–33. doi:10.1145/1243413.1243415

    work page doi:10.1145/1243413.1243415 2003

  55. [55]

    Klöckner, T

    A. Klöckner, T. Warburton, J. Bridge, J. Hesthaven, Nodal discontinuous Galerkin methods on graphics processors, Journal of Computational Physics 228 (2009) 7863–7882. doi: j.jcp.2009. 06.041

    2009

  56. [56]

    J. Chan, Z. Wang, A. Modave, J.-F. Remacle, T. Warbur- ton, GPU-accelerated discontinuous Galerkin methods on hy- brid meshes, Journal of Computational Physics 318 (2016) 142–

    2016

  57. [57]

    doi: https://doi.org/10.1016/j.jcp.2016.04.003

    work page doi:10.1016/j.jcp.2016.04.003 2016

  58. [58]

    Karakus, N

    A. Karakus, N. Chalmers, K. Świrydowicz, T. Warburton, A GPU accelerated discontinuous Galerkin incompressible flow solver, Journal of Computational Physics 390 (2019) 380–404. doi:https://doi.org/10.1016/j.jcp.2019.04.010

    work page doi:10.1016/j.jcp.2019.04.010 2019

  59. [59]

    Accessed: 2026-03-03

    NVIDIA, Nsight Compute™, https://developer.nvidia.com/ nsight-compute, 2026. Accessed: 2026-03-03

    2026

  60. [60]

    D. A. Kopriva, A conservative staggered-grid chebyshev mul- tidomain method for compressible flows. ii. a semi-structured method, Journal of computational physics 128 (1996) 475–488

    1996

  61. [61]

    P. J. Roache, Code verification by the method of manufactured solutions, J. Fluids Eng. 124 (2002) 4–10

    2002

  62. [62]

    D. J. Lusher, N. D. Sandham, Assessment of low-dissipative shock-capturing schemes for the compressible Taylor–Green vor- tex, AIAA Journal 59 (2021) 533–545

    2021

  63. [63]

    Sarkar, G

    S. Sarkar, G. Erlebacher, M. Y. Hussaini, H. O. Kreiss, The analysis and modelling of dilatational terms in compressible tur- bulence, Journal of Fluid Mechanics 227 (1991) 473–493

    1991

  64. [64]

    Sutherland, LII

    W. Sutherland, LII. The viscosity of gases and molecular force, The London, Edinburgh, and Dublin Philosophical Mag- azine and Journal of Science 36 (1893) 507–531. doi: 10.1080/ 14786449308620508

  65. [65]

    bg/index.html, 2026

    Discoverer HPC, Discoverer HPC docs, https://docs.discoverer. bg/index.html, 2026. Accessed: 2026-19-02

    2026

  66. [66]

    Accessed: 2026-19-02

    Höchstleistungsrechenzentrum Stuttgart (HLRS), Hunter (HPE), https://kb.hlrs.de/platforms/index.php/Hunter_ (HPE), 2026. Accessed: 2026-19-02

    2026

  67. [67]

    Accessed: 2026-19-02

    CINECA HPC, Leonardo, https://www.hpc.cineca.it/systems/ hardware/leonardo/, 2026. Accessed: 2026-19-02

    2026

  68. [68]

    Accessed: 2026-19-02

    Minho Advanced Computing Center (MACC), Deucalion User Guide, https://docs.macc.fccn.pt/, 2026. Accessed: 2026-19-02

    2026

  69. [69]

    Accessed: 2026-10- 04

    Oak Ridge National Laboratory (ORNL) Leadership Comput- ing Facility, Frontier User Guide, https://docs.olcf.ornl.gov/ systems/frontier_user_guide.html, 2026. Accessed: 2026-10- 04

    2026

  70. [70]

    Ac- cessed: 2026-12-03

    Jülich Supercomputing Centre (JSC), JUPITER - Exascale for Europe, https://www.fz-juelich.de/en/jsc/jupiter, 2026. Ac- cessed: 2026-12-03

    2026

  71. [71]

    Accessed: 2026-19-02

    IT4Innovations, Karolina, https://www.it4i.cz/en/ infrastructure/karolina, 2026. Accessed: 2026-19-02

    2026

  72. [72]

    Accessed: 2026- 19-02

    CSC - IT Center for Science, LUMI Documentation, https:// docs.lumi-supercomputer.eu/firststeps/, 2026. Accessed: 2026- 19-02

    2026

  73. [73]

    Accessed: 2026-19-02

    Barcelona Supercomputing Center (BSC), MareNostrum 5 Technical information, https://www.bsc.es/marenostrum/ marenostrum-5, 2026. Accessed: 2026-19-02

    2026

  74. [74]

    LuxProvide, MeluXina, https://www.luxprovide.lu/meluxina/,

  75. [75]

    Accessed: 2026-19-02

    2026

  76. [76]

    Accessed: 2026-19-02

    Institute for Information Science, Maribor (IZUM), Vega, https: //izum.si/en/vega-en/, 2026. Accessed: 2026-19-02

    2026

  77. [77]

    Advanced Micro Devices, AMD Instinct ™ MI250X Ac- celerators, https://www.amd.com/en/products/accelerators/ instinct/mi200/mi250x.html, 2026

    2026

  78. [78]

    com/, 2026

    SchedMD, Slurm Workload Manager, https://slurm.schedmd. com/, 2026. Accessed: 2026-27-01

    2026

  79. [79]

    Accessed: 2026-27-01

    Barcelona Supercomputing Center (BSC), EAR: Energy management framework for HPC, https://www.bsc.es/ 19 research-and-development/software-and-apps/software-list/ ear-energy-management-framework-hpc , 2026. Accessed: 2026-27-01

    2026

  80. [80]

    Kopper, M

    P. Kopper, M. P. Blind, A. Schwarz, M. Kurz, F. Rodach, S. M. Copplestone, A. D. Beck, PyHOPE: A Python Toolkit for Three-Dimensional Unstructured High-Order Meshes, Jour- nal of Open Source Software 10 (2025) 8769. URL: https: //doi.org/10.21105/joss.08769. doi: 10.21105/joss.08769

    work page doi:10.21105/joss.08769 2025

Showing first 80 references.