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arxiv: 2505.15912 · v2 · submitted 2025-05-21 · 🌀 gr-qc · astro-ph.IM

BHaHAHA: A Fast, Robust Apparent Horizon Finder Library for Numerical Relativity

Zachariah B. Etienne , Thiago Assump\c{c}\~ao , Leonardo Rosa Werneck , Samuel D. Tootle This is my paper

Pith reviewed 2026-05-22 13:29 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.IM
keywords apparent horizonsnumerical relativityhyperbolic relaxationmarginally outer trapped surfacesblack hole excisionopen source library
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The pith

A new library recasts the elliptic equation for apparent horizons as a damped nonlinear wave equation to speed up black hole detection in simulations.

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

The paper introduces BHaHAHA as an open-source library for finding apparent horizons that works independently of any specific numerical relativity code. It converts the usual elliptic partial differential equation into a damped nonlinear wave equation and adds multigrid-style refinement, over-relaxation, and OpenMP parallelization to reach convergence quickly. These changes produce large speed gains on single slices and make the library faster than AHFinderDirect when tracking horizons over many time steps on multiple cores. The method has been tested inside the Einstein Toolkit and BlackHoles@Home frameworks.

Core claim

BHaHAHA implements the first hyperbolic flow-based approach by recasting the elliptic PDE for a marginally outer trapped surface as a damped nonlinear wave equation, and with multigrid-inspired refinement, over-relaxation, and OpenMP achieves 64x speedups for difficult common-horizon finds and is approximately 2.1 times faster than AHFinderDirect for dynamic tracking at typical HPC core counts.

What carries the argument

Hyperbolic relaxation that converts the elliptic PDE for marginally outer trapped surfaces into a damped nonlinear wave equation, accelerated by multigrid-inspired refinement, over-relaxation, and OpenMP parallelization.

If this is right

  • Runtimes stay within 10 percent of single-core AHFinderDirect while beating it once multiple cores are used.
  • 64x faster common-horizon searches than a basic hyperbolic implementation on a single spacetime slice.
  • Approximately 2.1 times faster dynamic tracking than AHFinderDirect at accuracies set by host-metric interpolation.
  • Portable across different numerical relativity codes, as shown by successful use in both the Einstein Toolkit and BlackHoles@Home.

Where Pith is reading between the lines

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

  • The same damped-wave reformulation might speed up other elliptic problems that appear in general-relativity initial-data solvers.
  • Adding GPU kernels for the wave evolution step could push the speedup well beyond the current OpenMP gains on large clusters.
  • Faster horizon finding opens the possibility of updating excision regions more frequently during long binary-black-hole runs without extra cost.

Load-bearing premise

The hyperbolic relaxation method converges to the correct marginally outer trapped surface for all spacetimes and resolutions used in production runs without adding systematic bias beyond interpolation error in the input metric data.

What would settle it

Compare the horizon radius returned by BHaHAHA on a Schwarzschild or Kerr initial slice against the known analytic value, checking agreement to within the metric interpolation error reported by the host code.

Figures

Figures reproduced from arXiv: 2505.15912 by Leonardo Rosa Werneck, Samuel D. Tootle, Thiago Assump\c{c}\~ao, Zachariah B. Etienne.

Figure 1
Figure 1. Figure 1: Common apparent horizon (gray) for a q = 4 binary black hole configuration on a Brill–Lindquist initial data slice, with inner marginally trapped surfaces also plotted (green and blue spheroids). All surfaces were identified by BHaHAHA. While hyperbolic relaxation methods are generally easier to implement and often more robust than direct elliptic solvers—particularly when given poor initial guesses— they … view at source ↗
Figure 2
Figure 2. Figure 2: Acceleration Study. Convergence of the L∞ norm of the dimensionless expansion (||mscaleΘ||∞) versus the total number of gridpoint evaluations of Θ (GEs) for the q = 4 common horizon search starting from a poor initial guess located at the origin. Comparison of: baseline hyperbolic relaxation (No Accel, solid blue); an exponential fit to its late-time behavior (e −GE/(3.73×107 ) , dotted black); relaxation … view at source ↗
Figure 3
Figure 3. Figure 3: Top panel: Trajectory of the centroid of the smaller BH apparent horizon during the inspiral for GW150914, as tracked by BHaHAHA (BAH, solid blue line) and AHFinderDirect (AHFD, dashed orange line). Coordinates are scaled by the total mass M. Bottom panel: Absolute Euclidean distance ∆r/M between the centroids found by the two methods at corresponding simulation iterations, plotted against time t/M. For dy… view at source ↗
Figure 4
Figure 4. Figure 4: AH area comparison for the smaller BH in the GW150914 simulation. Top panel: Area evolution over time for three finder configurations: the control run using AHFinderDirect with 3rd-order Hermite interpolation (10−8 tolerance; dashed orange), AHFinderDirect using 4th-order Lagrange interpolation (10−8 tolerance; dotted green), and BHaHAHA (3rd-order Hermite interpolation, 10−5 tolerance; solid blue). Bottom… view at source ↗
Figure 5
Figure 5. Figure 5: GW150914-like BBH: First common horizon (gray) found, with inner marginally trapped surfaces plotted (green and blue spheroids). All surfaces identified by BHaHAHA. Relation to horizon pretracking. Schnetter, Herrmann, and Pollney introduced horizon pretracking as a complementary strategy for anticipating the formation of a common apparent horizon in BBH simulations [58, 23]. Rather than waiting for a MOTS… view at source ↗
Figure 6
Figure 6. Figure 6: Common apparent horizon (gray) for three equal-mass (m = 1) Brill￾Lindquist black holes, with inner horizons also shown (blue, red, green). The image depicts the system at the largest separation R for which a common horizon was identified (i.e., R = 1.1954995582) at a convergence tolerance of mscale||Θ||∞ = 3 × 10−8 (for total mass mscale = 3). All horizons were found using BHaHAHA implemented within the B… view at source ↗
read the original abstract

Apparent horizon (AH) finders are essential for characterizing black holes and excising their interiors in numerical relativity (NR) simulations. However, open-source AH finders to date are tightly coupled to individual NR codes. We introduce BHaHAHA, the BlackHoles@Home Apparent Horizon Algorithm, the first open-source, infrastructure-agnostic library for AH finding in NR. BHaHAHA implements the first-ever hyperbolic flow-based approach, recasting the elliptic partial differential equation for a marginally outer trapped surface as a damped nonlinear wave equation. To enhance performance, BHaHAHA incorporates a multigrid-inspired refinement strategy, an over-relaxation technique, and OpenMP parallelization. When compared to a na\"ive hyperbolic relaxation implementation, these enhancements result in 64x speedups for difficult common-horizon finds on a single spacetime slice, enabling BHaHAHA to achieve runtimes within 10% of the widely used (single-core) AHFinderDirect and outperform it on multiple cores. For dynamic horizon tracking with typical core counts on a high-performance-computing cluster, BHaHAHA is approximately 2.1 times faster than AHFinderDirect at accuracies limited by interpolation of metric data from the host NR code. Implemented and tested in both the Einstein Toolkit and BlackHoles@Home, BHaHAHA demonstrates that hyperbolic relaxation can be a robust, versatile, and performant approach for AH finding.

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

2 major / 2 minor

Summary. The paper introduces BHaHAHA, an open-source, infrastructure-agnostic library for apparent horizon finding in numerical relativity. It presents the first hyperbolic flow-based method that recasts the elliptic PDE for a marginally outer trapped surface as a damped nonlinear wave equation, augmented by multigrid-inspired refinement, over-relaxation, and OpenMP parallelization. Performance claims include 64x speedups over a naive hyperbolic implementation for common-horizon finds and approximately 2.1 times faster dynamic tracking than AHFinderDirect at typical HPC core counts, with runtimes within 10% of single-core AHFinderDirect; the library is tested in the Einstein Toolkit and BlackHoles@Home.

Significance. If the central performance and robustness claims hold, the work supplies a valuable, freely available AH finder that decouples from specific NR codes and demonstrates that hyperbolic relaxation can be competitive with established elliptic solvers. The open-source release, infrastructure independence, and concrete wall-clock comparisons constitute clear strengths for the NR community.

major comments (2)
  1. [Methods / Convergence tests] The manuscript asserts that the stationary state of the damped nonlinear wave equation solves the original elliptic MOTS condition, but does not report explicit checks that the final surface satisfies vanishing expansion (or the equivalent elliptic residual) to within the claimed accuracy, independent of damping coefficient and relaxation parameters. This verification is load-bearing for the robustness claim.
  2. [Results / Performance benchmarks] Table or figure reporting the 64x speedup and 2.1x dynamic-tracking comparison: the error budget relative to interpolation error from the host metric data is not quantified, nor is it shown that the hyperbolic method introduces no additional systematic offset beyond that interpolation error.
minor comments (2)
  1. [Results] Clarify the precise definition of 'common-horizon finds' and the range of spacetime slices and resolutions used in the single-slice timing tests.
  2. [Implementation] Add a brief statement on how the library handles the transition from initial guess to converged surface in highly dynamical spacetimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify important aspects of the method and its validation. We address each major comment below and have revised the manuscript to strengthen the presentation of the results.

read point-by-point responses

  1. Referee: [Methods / Convergence tests] The manuscript asserts that the stationary state of the damped nonlinear wave equation solves the original elliptic MOTS condition, but does not report explicit checks that the final surface satisfies vanishing expansion (or the equivalent elliptic residual) to within the claimed accuracy, independent of damping coefficient and relaxation parameters. This verification is load-bearing for the robustness claim.

    Authors: We agree that explicit numerical verification of the elliptic residual is necessary to support the robustness claim. While the equivalence follows from the derivation of the damped wave equation, we have added a new subsection to the Methods section in the revised manuscript. This subsection reports the computed expansion scalar (and equivalent elliptic residual) on the final surfaces for multiple values of the damping coefficient and over-relaxation parameter. The results confirm that the residual remains below the target tolerance across the tested parameter range and is insensitive to the specific choices, thereby verifying that the stationary state satisfies the MOTS condition to the claimed accuracy. revision: yes

  2. Referee: [Results / Performance benchmarks] Table or figure reporting the 64x speedup and 2.1x dynamic-tracking comparison: the error budget relative to interpolation error from the host metric data is not quantified, nor is it shown that the hyperbolic method introduces no additional systematic offset beyond that interpolation error.

    Authors: We acknowledge the importance of a quantified error budget. The manuscript already states that the reported accuracies are limited by interpolation of the metric data from the host NR code. In the revised Results section we have added an explicit quantification of this interpolation error (obtained by comparing interpolated fields against higher-resolution reference data) and a direct comparison of the hyperbolic solver output against a reference elliptic solution on the same interpolated data. These additions demonstrate that any additional systematic offset introduced by the hyperbolic method lies within the interpolation error and does not exceed it, with the relevant tables and figures updated to include error estimates and this comparison. revision: yes

Circularity Check

0 steps flagged

No circularity: implementation claims rest on external timings and benchmarks

full rationale

The paper describes a software library implementing a hyperbolic relaxation method by recasting an elliptic MOTS PDE into a damped nonlinear wave equation, with performance enhancements like multigrid refinement and OpenMP. Central claims are empirical speedups (64x vs naive, 2.1x vs AHFinderDirect) measured against independent codes and wall-clock data on production spacetimes. No equation reduces to a fitted parameter renamed as prediction, no load-bearing self-citation chain justifies the core method, and the stationary-state equivalence to the original elliptic condition is asserted via standard relaxation theory rather than by construction within the paper's own inputs. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical equivalence between the elliptic marginally-outer-trapped-surface equation and the damped nonlinear wave equation used for relaxation, plus standard assumptions about the smoothness of the metric data supplied by the host NR code. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The elliptic PDE for a marginally outer trapped surface can be recast as a damped nonlinear wave equation whose steady state solves the original problem.
    This equivalence is the foundation of the hyperbolic flow method and is invoked when the abstract states the recasting step.

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Works this paper leans on

62 extracted references · 62 canonical work pages · 22 internal anchors

  1. [1]

    Abbott B Pet al.(LIGO Scientific, Virgo) 2016Phys. Rev. Lett.116061102 (Preprint 1602.03837)

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Abbott T Det al.(LIGO Scientific, Virgo) 2016Phys. Rev. X6041014 (Preprint1606.01210)

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Abbott B Pet al.(LIGO Scientific, Virgo) 2019Phys. Rev. X9031040 (Preprint1811.12907)

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Abbott Ret al.(LIGO Scientific, Virgo) 2021Phys. Rev. X11021053 (Preprint2010.14527)

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Abbott Ret al.(LIGO Scientific, VIRGO) 2024Phys. Rev. D109022001 (Preprint 2108.01045)

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Abbott Ret al.(KAGRA, VIRGO, LIGO Scientific) 2023Phys. Rev. X13041039 (Preprint 2111.03606)

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    LIGO Scientific Collaboration and Virgo Collaboration (Virgo, LIGO Scientific) 2017Phys. Rev. Lett.119161101:1–18 (Preprint1710.05832)

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Abbott B Pet al.(LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, BHaHAHA: A Fast, Robust Apparent Horizon Finder Library for Numerical Relativity27 Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group...

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Penrose R 1965Phys. Rev. Lett.1457–59

    work page

  10. [10]

    Senovilla J M M 2011Int. J. Mod. Phys. D202139 (Preprint1107.1344)

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Thornburg J 2004Class. Quant. Grav.21743–766 (Preprintgr-qc/0306056)

    work page arXiv

  12. [12]

    Garfinkle D 2002Phys. Rev. D65044029 (Preprintgr-qc/0110013)

    work page arXiv

  13. [13]

    Quantum Grav.22425–451 (Preprintgr-qc/0407110)

    Pretorius F 2005Class. Quantum Grav.22425–451 (Preprintgr-qc/0407110)

    work page arXiv

  14. [14]

    Duez M D, Shapiro S L and Yo H J 2004Phys. Rev. D69104016 (Preprintgr-qc/0401076)

    work page arXiv

  15. [15]

    Hawke I, Loffler F and Nerozzi A 2005Phys. Rev. D71104006 (Preprintgr-qc/0501054)

    work page arXiv

  16. [16]

    Etienne Z B, Liu Y T, Paschalidis V and Shapiro S L 2012Phys. Rev. D85064029 (Preprint 1112.0568)

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    Zilh˜ ao M and L¨ offler F 2013Int. J. Mod. Phys. A281340014 (Preprint1305.5299)

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    Christodoulou D 1970Phys. Rev. Lett.251596–1597

    work page

  19. [19]

    Alcubierre Met al.2005Phys. Rev. D72044004 (Preprintgr-qc/0411149)

    work page arXiv

  20. [20]

    Smarr L 1973Phys. Rev. D7289–295

    work page

  21. [21]

    Brandt S R and Seidel E 1995Phys. Rev. D52870–886 (Preprintgr-qc/9412073)

    work page arXiv

  22. [22]

    Rel.710 (Preprintgr-qc/0407042)

    Ashtekar A and Krishnan B 2004Living Rev. Rel.710 (Preprintgr-qc/0407042)

    work page arXiv

  23. [23]

    Rel.103 (Preprintgr-qc/0512169)

    Thornburg J 2007Living Rev. Rel.103 (Preprintgr-qc/0512169)

    work page arXiv

  24. [24]

    thesis Queen Mary, U

    Fran¸ ca T 2023Binary Black Holes in Modified GravityPh.D. thesis Queen Mary, U. of London (main) (Preprint2308.12037)

    work page arXiv

  25. [25]

    Andrade et al.,GRChombo: An adaptable numerical relativity code for fundamental physics,J

    Andrade Tet al.2021J. Open Source Softw.63703 (Preprint2201.03458)

    work page arXiv

  26. [26]

    2025GRChomboGitHub repositoryhttps://github.com/GRTLCollaboration/GRChombo

    work page

  27. [27]

    Hui H K and Lin L M 2025Class. Quant. Grav.42055008 (Preprint2404.16511)

    work page arXiv

  28. [28]

    Gundlach C 1998Phys. Rev. D57863–875 (Preprintgr-qc/9707050)

    work page arXiv

  29. [29]

    Fernando M, Neilsen D, Zlochower Y, Hirschmann E W and Sundar H 2023Phys. Rev. D107 064035 (Preprint2211.11575)

    work page arXiv

  30. [30]

    2025Dendro-GRGitHub repositoryhttps://github.com/paralab/Dendro-GR

    work page

  31. [31]

    Kidder L E, Field S E, Foucart F, Schnetter E, Teukolsky S A, Bohn A, Deppe N, Diener P, H´ ebert F, Lippuner J, Miller J, Ott C D, Scheel M A and Vincent T 2017J. Comp. Phys.33584–114 (Preprint1609.00098)

    work page arXiv

  32. [32]

    2025SpECTREGitHub repositoryhttps://github.com/sxs-collaboration/spectre

    work page

  33. [33]

    Alcubierre M, Brandt S, Bruegmann B, Gundlach C, Masso J, Seidel E and Walker P 2000Class. Quant. Grav.172159–2190 (Preprintgr-qc/9809004)

    work page arXiv

  34. [34]

    The Einstein Toolkit: A Community Computational Infrastructure for Relativistic Astrophysics

    L¨ offler Fet al.2011Class. Quantum Grav.29115001:1–56 (PreprintarXiv:1111.3344[gr-qc])

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    R¨ uter H R, Hilditch D, Bugner M and Br¨ ugmann B 2018Phys. Rev. D98084044 (Preprint 1708.07358)

    work page internal anchor Pith review Pith/arXiv arXiv

  36. [36]

    Assumpcao T, Werneck L R, Jacques T P and Etienne Z B 2022Phys. Rev. D105104037 (Preprint 2111.02424)

    work page arXiv

  37. [37]

    Tootle S D, Werneck L R, Assump¸ c˜ ao T, Jacques T P and Etienne Z B 2025Class. Quant. Grav. 42115005 (Preprint2501.14030)

    work page arXiv

  38. [38]

    Alcubierre M and Bruegmann B 2001Phys. Rev. D63104006 (Preprintgr-qc/0008067)

    work page arXiv

  39. [39]

    Campanelli M, Lousto C O, Marronetti P and Zlochower Y 2006Phys. Rev. Lett.96111101 (Preprintgr-qc/0511048)

    work page arXiv

  40. [40]

    Brown J D 2009Phys. Rev. D80084042 (Preprint0908.3814) BHaHAHA: A Fast, Robust Apparent Horizon Finder Library for Numerical Relativity28

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    Montero P J and Cordero-Carrion I 2012Phys. Rev. D85124037:1–10 (Preprint1204.5377)

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [42]

    Baumgarte T W, Montero P J, Cordero-Carrion I and M¨ uller E 2013Phys. Rev. D87044026:1–14 (Preprint1211.6632)

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    2025NRPyGitHub repositoryhttps://github.com/nrpy/nrpy

    work page

  44. [44]

    Ruchlin I, Etienne Z B and Baumgarte T W 2018Phys. Rev. D97064036 (Preprint arXiv: 1712.07658[gr-qc])

    work page internal anchor Pith review Pith/arXiv arXiv

  45. [45]

    Lin L M and Novak J 2007Class. Quant. Grav.242665–2676 (Preprintgr-qc/0702038)

    work page arXiv

  46. [46]

    Arnowitt R L, Deser S and Misner C W 2008Gen. Rel. Grav.401997–2027 (Preprint gr-qc/0405109)

    work page internal anchor Pith review Pith/arXiv arXiv 2027

  47. [47]

    Etienne Z B 2024Phys. Rev. D110064045 (Preprint2404.01137)

    work page arXiv

  48. [48]

    Meurer A, Smith C P, Paprocki M, ˇCert´ ık O, Kirpichev S B, Rocklin M, Kumar A, Ivanov S, Moore J K, Singh S, Rathnayake T, Vig S, Granger B E, Muller R P, Bonazzi F, Gupta H, Vats S, Johansson F, Pedregosa F, Curry M J, Terrel A R, Rouˇ cka v, Saboo A, Fernando I, Kulal S, Cimrman R and Scopatz A 2017PeerJ Computer Science3e103:1–27 ISSN 2376-5992 URL h...

    work page doi:10.7717/peerj-cs.103

  49. [49]

    org/stable/3649684

    Gottlieb S, Shu C W and Tadmor E 2001SIAM Review4389–112 URL https://www.jstor. org/stable/3649684

    work page arXiv

  50. [50]

    Rev.131471–476

    Brill D R and Lindquist R W 1963Phys. Rev.131471–476

    work page

  51. [51]

    Evolutions in 3D numerical relativity using fixed mesh refinement

    Schnetter E, Hawley S H and Hawke I 2004Class. Quantum Grav.211465–1488 (Preprint gr-qc/0310042)

    work page internal anchor Pith review Pith/arXiv arXiv

  52. [52]

    2024 Carpet: Adaptive mesh refinement for the Cactus Framework http://www.carpetcode.org/

    work page 2024

  53. [53]

    Wardell B, Hinder I and Bentivegna E 2016 Simulation of GW150914 binary black hole merger using the Einstein Toolkit URLhttps://doi.org/10.5281/zenodo.155394

    work page doi:10.5281/zenodo.155394 2016

  54. [54]

    2024 Einstein Toolkit Consortium Homepagehttp://einsteintoolkit.org/

    work page 2024

  55. [55]

    Abbott B Pet al.(LIGO Scientific, Virgo) 2016Phys. Rev. Lett.116241102 (Preprint 1602.03840)

    work page internal anchor Pith review Pith/arXiv arXiv

  56. [56]

    Ansorg M, Br¨ ugmann B and Tichy W 2004Phys. Rev. D70064011 (Preprintgr-qc/0404056)

    work page arXiv

  57. [57]

    Etienne Z 2024 Updated BaikalVacuum with all improvements implemented available at https: //github.com/zachetienne/baikalvacimproved

    work page 2024

  58. [58]

    Schnetter E, Herrmann F and Pollney D 2005Phys. Rev. D71044033 (Preprint gr-qc/0410081)

    work page internal anchor Pith review Pith/arXiv arXiv

  59. [59]

    Ashtekar A and Krishnan B 2002Phys. Rev. Lett.89261101 (Preprintgr-qc/0207080)

    work page arXiv

  60. [60]

    Ashtekar A and Krishnan B 2003Phys. Rev. D68104030 (Preprintgr-qc/0308033)

    work page arXiv

  61. [61]

    Silberman Z J, Adams T R, Faber J A, Etienne Z B and Ruchlin I 2019J. Comput. Phys.379 421–437 (Preprint1803.10207)

    work page internal anchor Pith review Pith/arXiv arXiv

  62. [62]

    Idaho C3+3 Collaboration 2022 Falcon: High Performance Supercomputer DOI: 10.7923/falcon.id

    work page doi:10.7923/falcon.id 2022