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REVIEW 3 major objections 5 minor 84 references

HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A directive-based OpenACC port of the HORSES3D discontinuous Galerkin solver scales near-ideally to 2,048 GPUs and reaches 10.7 billion degrees of freedom on a realistic high-lift wing-body configuration.

desk verdict Honest, workmanlike OpenACC port with a real scaling claim that at the flagship 2048-GPU point is partly an extrapolation from lower-order data. read the letter →

arxiv 2607.26674 v1 pith:CHV72EF3 submitted 2026-07-29 math.NA cs.NA

Gerasimos Ntoukas , Gonzalo Rubio , Abbas Ballout , Stefano Colombo , David Huergo , Eduardo Jané , Albert Jiménez-Ramos , Hatem Kessasra
show 8 more authors
Himpu Marbona Oscar Mariño Rodrigo Salado Pol Solé-Miró Andrés M. Rueda-Ramírez Miguel Chávez-Módena Eusebio Valero Esteban Ferrer
This is my paper · ORCID
classification math.NAcs.NA MSC 65M6065Y0576M1076F65
keywords high-ordermethodsdiscontinuousGalerkinGPUaccelerationOpenACCmulti-GPUscalingcompressibleNavier-Stokesequationsbillion-degree-of-freedomsimulationsDGSEM
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper reports a GPU port of HORSES3D, an open-source high-order discontinuous Galerkin solver for compressible flow, achieved by adding OpenACC directives to the existing Fortran code rather than rewriting it. The port maps each mesh element to a GPU thread block and nodal operations to vector lanes, keeping data resident on the GPU. On a canonical transition-to-turbulence benchmark and on a realistic high-lift wing-body configuration with up to 20.8 million hexahedral elements, the solver shows near-ideal strong and weak scaling up to 2,048 GPUs, with the largest run reaching about 10.7 billion degrees of freedom. The central practical claim is that performance is controlled by the number of elements per GPU: once each device holds roughly 16,000–20,000 elements, efficiency saturates and scaling becomes near-linear, even for complex geometries.

What carries the argument

The central mechanism is the element-local structure of nodal DGSEM, exploited by mapping the element loop to OpenACC gangs (thread blocks) and the nodal quadrature loop to vector lanes, with register-resident accumulation to avoid global-memory round trips. For split-form two-point-flux schemes, the paper deliberately abandons CPU-side flux-symmetry reuse: because scattering one flux to two nodes would require atomic updates across lanes, it recomputes each two-point flux for both orderings and keeps accumulation private per lane, fusing advective and viscous contributions into one kernel. The measured 'elements per GPU' threshold then acts as a universal load parameter that predicts when c

What would settle it

Take a benchmark case that fits on one GPU, such as 16,384 hexahedral elements at polynomial order 5, measure the efficiency metric for 1, 2, 4, 8, up to 256 GPUs, and compute strong-scaling efficiency using the single-GPU run as the reference; if efficiency at 256 GPUs is below about 80% even though each GPU still has more than 16,000 elements, then the near-ideal scaling reported in the paper is an artifact of measuring relative to a multi-GPU baseline rather than to a single GPU.

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Extended reading notes

Core claim

The discovery is that a high-order DGSEM CFD code can be made GPU-resident by a relatively low-touch OpenACC port, without changing the numerical formulation, and still scale to thousands of GPUs. The key evidence is that solver efficiency, measured as wall-time per degree of freedom per Runge-Kutta stage, improves with polynomial order, saturates when the load exceeds about 16,000 to 20,000 elements per GPU, and remains within a factor of roughly two of single-GPU performance on an industrial high-lift wing-body configuration. At 2,048 GPUs with polynomial order 7 and a 20.8-million-element mesh, the solver sustains strong-scaling efficiencies around 80% or better while the load stays above

Load-bearing premise

The claim of near-ideal scaling rests on assuming that the smallest multi-GPU run used as the speedup baseline already achieves perfect parallel efficiency; if that baseline carries significant communication or load-imbalance costs, reported speedups overstate true scalability.

Editorial extensions

If this is right

  • At polynomial orders 5–7 and loads above about 20,000 elements per GPU, near-ideal strong scaling is maintained up to roughly 1,024 GPUs; 2,048 GPUs remain usable with lower efficiency.
  • Higher polynomial order yields better GPU efficiency for a fixed number of degrees of freedom, so p-refinement is computationally cheaper than h-refinement on this architecture.
  • The same OpenACC codebase can target CPU and GPU with identical numerical results, verified by manufactured-solution convergence and turbulent-flow validation.
  • The elements-per-GPU threshold is a practical configuration guideline: below roughly 16,000 elements per device, communication dominates and efficiency degrades sharply.
  • GPU-resident execution, with all data on the device and minimal host-device transfers, is sufficient to reach billion-degree-of-freedom industrial-scale LES within current HPC allocations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because strong-scaling baselines are multi-GPU runs, true single-GPU-relative speedups could be lower than reported; measuring against a single GPU would yield the honest efficiency curve.
  • The elements-per-GPU rule could be turned into a portable cost model: runtime per step might be predicted from element count divided by GPU count, independent of total problem size, which would help scheduler-aware resource allocation.
  • The threshold is likely to shift with hardware generations: accelerators with more memory bandwidth or more compute per device may lower or raise it, and the same methodology could re-measure it on other architectures.
  • The paper's positioning as a data engine for AI-era CFD would be strengthened by a throughput comparison against CPU baselines, including energy cost; the paper reports efficiency metric but not power or CPU speedup.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents HORSES3D-GPU, an OpenACC-based multi-GPU implementation of the high-order discontinuous Galerkin solver HORSES3D. It verifies the port with a manufactured-solutions convergence study and validates it on the Taylor–Green vortex and a wall-modeled turbulent channel flow. Performance is assessed on MareNostrum 5's NVIDIA H100 partition. For the TGV benchmark, the authors report single-GPU PID data, identify a practical regime of 16,000–20,000 elements per GPU, and show strong/weak scaling on up to 256 GPUs. For the CRM-HL wing–body configuration, they report PID across four meshes and polynomial orders P=1–7, strong scaling on up to 2048 GPUs, and a flagship simulation of the h3 mesh at P=7 with approximately 10.7e9 DOF. The central claims are that HORSES3D preserves its single-GPU performance characteristics on industrial geometries and scales efficiently to 2048 GPUs.

Significance. If fully established, the paper would be a useful contribution to GPU-based high-order CFD: it demonstrates that an OpenACC port can retain a single Fortran source, provides an open-source implementation, uses a clearly defined performance metric, and reports data on a production-scale high-lift configuration. The authors are also explicit about several limitations, including the multi-GPU baseline normalization and the memory-bound nature of the kernels. However, the headline scaling claim rests on a data gap: the 2048-GPU P=7 flagship run is mentioned but no measured speedup or efficiency is reported for it, and the CRM strong-scaling curves are normalized to a multi-GPU baseline, so the reported percentages are not absolute parallel efficiencies.

major comments (3)
  1. [Section 5.2.2 / Figure 11] The claimed 2048-GPU result for the largest simulation is not present in the scaling data. The h3 panel of Figure 11 shows strong-scaling curves only for P=1, P=2, and P=3, while the text states that the h3/P=7/2048 run (≈10.7e9 DOF) was performed. No wall-time, speedup, PID, or efficiency is reported for this run; the only 2048-GPU efficiency given is about 53% for P=2. Consequently, the abstract's statement that the solver 'scale[s] efficiently to 2,048 GPUs' is an extrapolation from lower-order curves rather than a directly measured property of the flagship run. Please either report the timing/speedup/PID for the h3/P=7/2048 point or restrict the scaling claim to the polynomial orders actually shown and describe the P=7/2048 run as a capability demonstration.
  2. [Section 5.2.1 / Eq. (10)] The strong-scaling speedups in Figure 11 are normalized to the smallest multi-GPU configuration that can fit each mesh, so the leftmost point is 100% by construction and any communication or load-imbalance overhead present at the baseline is invisible. The paper acknowledges this in Section 5.2.1, but Section 5.2.2 then interprets the ratios as 'efficiency' (e.g., 53% at 2048 GPUs for h3/P=2). For the CRM case there is no single-GPU anchor; the TGV single-GPU comparison in Section 4.2.2 is for a structured periodic mesh and cannot be transferred directly to the unstructured CRM with complex boundary treatment. Please provide a single-GPU-equivalent reference for CRM, or an independent estimate of the baseline parallel efficiency, or consistently label the values as 'speedup relative to the multi-GPU baseline' rather than as absolute efficiency.
  3. [Section 5.2.1 / Figure 10] The comparison of CRM PID values with the single-GPU TGV PID values is used to support the claim that the solver 'preserves its performance characteristics.' Because the CRM PID is obtained from multi-GPU runs and rescaled under an ideal-strong-scaling assumption, it cannot separate the cost of complex boundary treatment from parallel-efficiency losses due to MPI communication, load imbalance, or network contention. The close agreement between CRM and TGV PID therefore establishes only that the combined effect remains comparable, not that the single-GPU performance characteristics are preserved. Please state this inferential limitation explicitly when drawing this conclusion.
minor comments (5)
  1. [Section 4.2.2 / Figure 7] The weak-scaling baseline is described as a single 4-GPU node, yet the text and figure also include points for 1 and 2 GPUs and report efficiencies above 1 for those points. Please clarify the reference configuration and state whether the problem size is scaled from 1 GPU or from 4 GPUs.
  2. [Section 3.1 / Figure 1] The opening sentence says the figure shows 'exponential convergence, for h- and p-refinement.' h-refinement with fixed polynomial order gives algebraic convergence; the following sentences correctly distinguish the two behaviors. The opening should be reworded to avoid this imprecision.
  3. [Appendix B] The wall-model appendix states that in some MPI configurations the wall-model input may correspond to the previous Runge–Kutta stage, 'at the cost of a slight temporal lag.' This is an acknowledged approximation in the wall-modeled LES validation; please quantify the lag or at least discuss its expected effect on the reported mean-velocity profiles.
  4. [Sections 4.2 and 5.2] The performance measurements appear to be single runs on a shared HPC system, with no repeated runs or variance estimates. A brief statement about run-to-run variability and whether the reported times are representative would improve reproducibility.
  5. [Figure 11 / Section 5.2.2] Some efficiency percentages quoted in the text do not obviously match the annotations in Figure 11. For example, the text reports 56% at 1024 GPUs for h1/P=4, while the figure panel appears to contain different annotation values. Please reconcile the text and figure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: empirical performance study with external validation; baseline-relative normalization is a caveat, not a circular reduction.

full rationale

This paper is an empirical performance study rather than a derivation of a theory from its own output. The central scalability claims rest on direct wall-clock measurements: the PID metric (Eq. 10) is defined from measured wall time, number of ranks, RK stages, DOF, and time steps, and the strong/weak scaling curves plot measured data. The '16,000–20,000 elements per GPU' threshold is a descriptive summary of the measured PID curves (Figs. 4, 5, 9), not a parameter fitted to one subset of data and then used to predict the same subset. Verification and validation are anchored to external references: MMS convergence tests; Taylor–Green vortex compared with Bull and Jameson [78] and Carton de Wiart et al. [79]; turbulent channel flow compared with Lee and Moser [82]. The GPU port's numerical content is therefore checked against independent results, not assumed from self-citations. Self-citations appear for background and implementation details ([4]) and for the choice of KEP/split-form configurations ([43], [77], [83], [84]), but none of these citations carries the central large-scale performance claim; the scaling result is measured directly on the target machine. The only notable caveat is methodological rather than circular: Section 5.2.1 states that CRM measurements are multi-GPU only and are 'rescaled under the assumption of ideal strong scalability,' and Fig. 11 is normalized to the smallest number of GPUs that fits each mesh. Additionally, the flagship h3/P=7/2048-GPU run is mentioned in Section 5.2.2 but does not appear in the plotted strong-scaling curves. This limits the strength of the 'scales efficiently to 2048 GPUs' statement from the displayed data, but it is a normalization/completeness limitation, not an equation, fitted parameter, or self-citation chain that reduces the claim to its own inputs by construction. No circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims are empirical performance results; the paper contributes no new mathematical derivations. It relies on the correctness of the original CPU solver and on the accuracy of external reference data for validation. The scaling threshold is an observed regularity, not a fitted parameter.

assumptions (4)
  • domain assumption The CPU HORSES3D solver (Ferrer et al. 2023, Ref [4]) is numerically correct and validated.
    The paper relies on the CPU solver as the reference for numerical formulation; the GPU port is verified by matching CPU results (Section 3). If the CPU solver had undiscovered errors, the GPU port's correctness claim would inherit them.
  • standard math DGSEM exhibits the expected p/h convergence rates for the smooth manufactured solution.
    Section 3.1 relies on theoretical convergence order of DGSEM to interpret MMS results; no independent exact solution is used beyond the manufactured solution.
  • domain assumption Reference datasets (Bull & Jameson TGV, Lee & Moser channel DNS, Carton de Wiart spectrum) are accurate and appropriate for comparison.
    Validation in Sections 3.2–3.3 is qualitative agreement with these references; the paper does not quantify the uncertainty of the references.
  • domain assumption The MareNostrum 5 hardware behaves as specified (H100 HBM2e bandwidth, NVLink, InfiniBand) during the runs.
    Performance conclusions depend on no hardware anomalies; the paper notes the HBM2e limitation as a relevant factor (Section 4.1).

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Cite this review

Pith. "Pith review of HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems." pith.science (2026). https://pith.science/paper/CHV72EF3

@misc{pith2026260726674,
  author       = {Pith},
  title        = {Pith review of: HORSES3D-GPU: A high-order discontinuous Galerkin solver for multi-GPU systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CHV72EF3}},
  note         = {Machine review of arXiv:2607.26674}
}
abstract

We present the GPU acceleration and large-scale performance assessment of HORSES3D, an open-source high-order discontinuous Galerkin solver for computational fluid dynamics. The solver is ported to NVIDIA GPU architectures using OpenACC directives, preserving the original Fortran code structure while enabling GPU-resident execution of the main computational kernels. The implementation exploits the element-local structure of discontinuous Galerkin spectral element methods by mapping element-level loops to GPU gangs and nodal operations to vector-level parallelism. The GPU version is verified using the method of manufactured solutions and validated on canonical turbulent-flow benchmarks. Its performance is assessed on the MareNostrum 5 accelerated partition using NVIDIA H100 GPUs. Taylor-Green vortex benchmarks show that solver efficiency improves with polynomial order and that near-ideal strong and weak scaling is obtained when the workload exceeds approximately 16,000 to 20,000 elements per GPU. The solver is further evaluated on the High-Lift Common Research Model wing-body configuration, which involves a complex geometry, realistic boundary conditions, and unstructured meshes with up to 20.8 million hexahedral elements. Simulations with polynomial orders up to $P=7$ reach approximately $10.7 \times 10^9$ degrees of freedom and scale efficiently to 2,048 GPUs. The results demonstrate that HORSES3D preserves its performance characteristics for industrially relevant configurations and can exploit modern GPU-based supercomputers for billion-degree-of-freedom high-order CFD simulations.

Figures

Figures reproduced from arXiv: 2607.26674 by the authors.

Figure 1
Figure 1. Convergence of the L 2 error against the number of degrees of freedom for different meshes and polynomial orders in the manufactured solution test case. analytical solution. The compressible Navier–Stokes equations are solved, for a Mach number of M = 0.3 and a Reynolds number of Re = 100000, using the DGSEM formulation with Gauss–Lobatto nodes and the Roe Riemann solver for inter-element fluxes. Viscous terms are d… view at source ↗
Figure 2
Figure 2. Validation of the GPU-accelerated HORSES3D solver for the Taylor–Green vortex at Re = 1600. Panel (a) shows the kinetic energy dissipation rate compared with the high-resolution DRP reference solution of Bull and Jameson [78]. Panel (b) shows the kinetic energy spectrum at t/tc = 9 compared with the pseudo-spectral reference solution of Carton de Wiart et al. [79]. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Mean velocity profiles for the turbulent channel flow test case using wall function. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Single GPU performance of HORSES3D for the KEP and SDG schemes for different polynomial orders and different mesh sizes. The figure depicts performance index, PID, (Eq. 10, the lower the better) vs degrees of freedom (DOF) per GPU. The second important result of [PITH…
Figure 5
Figure 5. Figure 5: Strong scalability for the KEP scheme for polynomial orders [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Strong scalability results for the KEP and SDG schemes for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Weak scalability of HORSES3D for P = 5 and 16384 elements per GPU. Results are relative to single node performance (4 GPUs). 5.1 Computational Setup The simulations are performed using HORSES3D and the compressible Navier–Stokes formulation, employing Chandrashekar ent…
Figure 8
Figure 8. Figure 8: Representative flow solution used in the performance assessment. Instantaneous pres [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Performance index PID (Eq. (10)) for the CRM wing–body configuration across mesh levels h0–h3 and polynomial orders P = 1–7. The results are shown as a function of elements per GPU, highlighting the onset of efficiency saturation once a minimum load of approximately 2 …
Figure 10
Figure 10. Figure 10: Performance index PID (Eq. (10)) for the CRM wing–body configuration across mesh levels h0–h3 and polynomial orders P = 1–7. The results are reported as a function of degrees of freedom per GPU, allowing direct assessment of efficiency saturation with increasing load …
Figure 11
Figure 11. Figure 11: Strong scaling speedup relative to the baseline GPU count for the CRM wing–body [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Weak scaling behaviour of the CRM wing–body configuration grouped by clusters [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.