# Modeling the Dynamics of Heavy-Ion Collisions with a Hydrodynamic Model Using a Graphics Processor

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## Abstract

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## 1. Introduction

## 2. Methodology

## 3. Results

#### 3.1. Initial Conditions

#### 3.2. Hypersurface Extraction Methods

#### 3.3. Hydrodynamics and Kinematic Description

#### 3.4. Simulation of Pb-Pb Collisions at $\sqrt{{s}_{NN}}=2.76$ TeV

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kliemant, M.; Sahoo, R.; Schuster, T.; Stock, R. Global Properties of Nucleus-Nucleus Collisions. In The Physics of the Quark-Gluon Plasma; Springer: Berlin/Heidelberg, Germany, 2010; Volume 785, pp. 23–103. [Google Scholar] [CrossRef] [Green Version]
- Satz, H. The Thermodynamics of Quarks and Gluons. Lect. Notes Phys.
**2010**, 785, 1–21. [Google Scholar] [CrossRef] [Green Version] - Hirano, T.; van der Kolk, N.; Bilandzic, A. Hydrodynamics and Flow. Lect. Notes Phys.
**2010**, 785, 139–178. [Google Scholar] [CrossRef] [Green Version] - Schenke, B.; Jeon, S.; Gale, C. (3 + 1) D hydrodynamic simulation of relativistic heavy-ion collisions. Phys. Rev. C
**2010**, 82, 014903. [Google Scholar] [CrossRef] - Pang, L.G.; Petersen, H.; Wang, X.N. Pseudorapidity distribution and decorrelation of anisotropic flow within CLVisc hydrodynamics. Phys. Rev. C
**2018**, 97, 064918. [Google Scholar] [CrossRef] [Green Version] - Petersen, H.; Steinheimer, J.; Burau, G.; Bleicher, M.; Stöcker, H. Fully integrated transport approach to heavy ion reactions with an intermediate hydrodynamic stage. Phys. Rev. C
**2008**, 78, 044901. [Google Scholar] [CrossRef] - Gerhard, J.; Lindenstruth, V.; Bleicher, M. Relativistic hydrodynamics on graphic cards. Comput. Phys. Commun.
**2013**, 184, 311–319. [Google Scholar] [CrossRef] [Green Version] - Karpenko, I.; Huovinen, P.; Bleicher, M. A 3 + 1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions. Comput. Phys. Commun.
**2014**, 185, 3016–3027. [Google Scholar] [CrossRef] [Green Version] - Alqahtani, M.; Nopoush, M.; Ryblewski, R.; Strickland, M. (3+ 1) D quasiparticle anisotropic hydrodynamics for ultrarelativistic heavy-ion collisions. Phys. Rev. Lett.
**2017**, 119, 042301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lin, Z.W.; Ko, C.M.; Li, B.A.; Zhang, B.; Pal, S. Multiphase transport model for relativistic heavy ion collisions. Phys. Rev. C
**2005**, 72, 064901. [Google Scholar] [CrossRef] [Green Version] - Schenke, B.; Tribedy, P.; Venugopalan, R. Fluctuating Glasma initial conditions and flow in heavy ion collisions. Phys. Rev. Lett.
**2012**, 108, 252301. [Google Scholar] [CrossRef] - Moreland, J.S.; Bernhard, J.E.; Bass, S.A. Alternative ansatz to wounded nucleon and binary collision scaling in high-energy nuclear collisions. Phys. Rev. C
**2015**, 92, 011901. [Google Scholar] [CrossRef] [Green Version] - Donoho, D.L.; Grimes, C. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. USA
**2003**, 100, 5591–5596. [Google Scholar] [CrossRef] [Green Version] - Boris, J.P.; Book, D.L. Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys.
**1973**, 11, 38–69. [Google Scholar] [CrossRef] - Kraposhin, M.; Bovtrikova, A.; Strijhak, S. Adaptation of Kurganov-Tadmor numerical scheme for applying in combination with the PISO method in numerical simulation of flows in a wide range of Mach numbers. Procedia Comput. Sci.
**2015**, 66, 43–52. [Google Scholar] [CrossRef] [Green Version] - Huovinen, P.; Petersen, H. Particlization in hybrid models. Eur. Phys. J. A
**2012**, 48, 171. [Google Scholar] [CrossRef] [Green Version] - Weil, J.; Steinberg, V.; Staudenmaier, J.; Pang, L.; Oliinychenko, D.; Mohs, J.; Kretz, M.; Kehrenberg, T.; Goldschmidt, A.; Bäuchle, B.; et al. Particle production and equilibrium properties within a new hadron transport approach for heavy-ion collisions. Phys. Rev. C
**2016**, 94, 054905. [Google Scholar] [CrossRef] - Pierog, T.; Werner, K. EPOS model and ultra high energy cosmic rays. Nucl. Phys. B-Proc. Suppl.
**2009**, 196, 102–105. [Google Scholar] [CrossRef] [Green Version] - Bleicher, M.; Zabrodin, E.; Spieles, C.; Bass, S.A.; Ernst, C.; Soff, S.; Bravina, L.; Belkacem, M.; Weber, H.; Stöcker, H.; et al. Relativistic hadron-hadron collisions in the ultra-relativistic quantum molecular dynamics model. J. Phys. G Nucl. Part. Phys.
**1999**, 25, 1859. [Google Scholar] [CrossRef] - Miller, M.L.; Reygers, K.; Sanders, S.J.; Steinberg, P. Glauber modeling in high energy nuclear collisions. Ann. Rev. Nucl. Part. Sci.
**2007**, 57, 205–243. [Google Scholar] [CrossRef] [Green Version] - Gelis, F.; Iancu, E.; Jalilian-Marian, J.; Venugopalan, R. The Color Glass Condensate. Ann. Rev. Nucl. Part. Sci.
**2010**, 60, 463–489. [Google Scholar] [CrossRef] [Green Version] - Słodkowski, M.; Marcinkowski, P.; Gawryszewski, P.; Kikoła, D.; Porter-Sobieraj, J. Modeling of modifications induced by jets in the relativistic bulk nuclear matter. J. Phys. Conf. Ser.
**2018**, 1085, 052001. [Google Scholar] [CrossRef] - Słodkowski, M.; Gawryszewski, P.; Marcinkowski, P.; Setniewski, D.; Porter-Sobieraj, J. Simulations of Energy Losses in the Bulk Nuclear Medium Using Hydrodynamics on the Graphics Cards (GPU). Proceedings
**2019**, 10, 27. [Google Scholar] [CrossRef] [Green Version] - Marcinkowski, P.; Słodkowski, M.; Kikoła, D.; Gawryszewski, P.; Sikorski, J.; Porter-Sobieraj, J.; Zygmunt, B. Jet-induced modifications of the characteristic of the bulk nuclear matter. In Proceedings of the 15th International Conference on Strangeness in Quark Matter (SQM2015), Dubna, Russia, 6–11 July 2015; Volume 668, p. 012115. [Google Scholar] [CrossRef] [Green Version]
- Porter-Sobieraj, J.; Słodkowski, M.; Kikoła, D.; Sikorski, J.; Aszklar, P. A MUSTA-FORCE algorithm for solving partial differential equations of relativistic hydrodynamics. Int. J. Nonlinear Sci. Numer. Simul.
**2018**, 19, 25–35. [Google Scholar] [CrossRef] - Porter-Sobieraj, J.; Cygert, S.; Kikoła, D.; Sikorski, J.; Słodkowski, M. Optimizing the computation of a parallel 3D finite difference algorithm for graphics processing units. Concurr. Comput. Pract. Exp.
**2015**, 27, 1591–1602. [Google Scholar] [CrossRef] - Nvidia, CUDA C++ Programming Guide; PG-02829-001_v11.2; Nvidia Corporation. Available online: https://docs.nvidia.com/cuda/pdf/CUDA_C_Programming_Guide.pdf (accessed on 19 March 2020).
- Słodkowski, M.; Gawryszewski, P.; Setniewski, D. Study of the influence of initial-state fluctuations on hydrodynamic simulations. EPJ Web Conf.
**2020**, 245, 06005. [Google Scholar] [CrossRef] - Pratt, S. Accounting for backflow in hydrodynamic-Boltzmann interfaces. Phys. Rev. C
**2014**, 89, 024910. [Google Scholar] [CrossRef] [Green Version] - Adam, J.; Adamová, D.; Aggarwal, M.M.; Rinella, G.A.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S.; Aiola, S.; et al. Anisotropic flow of charged particles in Pb-Pb collisions at s N N= 5.02 TeV. Phys. Rev. Lett.
**2016**, 116, 132302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Romatschke, P. New developments in relativistic viscous hydrodynamics. Int. J. Mod. Phys. E
**2010**, 19, 1–53. [Google Scholar] [CrossRef] - Romatschke, P.; Romatschke, U. Relativistic Fluid Dynamics in and out of Equilibrium: And Applications to Relativistic Nuclear Collisions; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Burrage, K.; Butcher, J.C.; Chipman, F. An implementation of singly-implicit Runge-Kutta methods. BIT Numer. Math.
**1980**, 20, 326–340. [Google Scholar] [CrossRef] - Liu, X.D.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys.
**1994**, 115, 200–212. [Google Scholar] [CrossRef] [Green Version] - Cooper, F.; Frye, G. Comment on the Single Particle Distribution in the Hydrodynamic and Statistical Thermodynamic Models of Multiparticle Production. Phys. Rev. D
**1974**, 10, 186. [Google Scholar] [CrossRef] - Huovinen, P.; Holopainen, H. Cornelius—User’s Guide. Available online: https://itp.uni-frankfurt.de/~huovinen/cornelius/guide.pdf (accessed on 19 March 2020).
- Lorensen, W.; Cline, H. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. ACM SIGGRAPH Comput. Graph.
**1987**, 21, 163–169. [Google Scholar] [CrossRef] - Nielson, G.M.; Hamann, B. The asymptotic decider: Resolving the ambiguity in Marching Cubes. IEEE Vis.
**1991**, 91, 83–91. [Google Scholar] [CrossRef] - Montani, C.; Scateni, R.; Scopigno, R. A modified look-up table for implicit disambiguation of Marching Cubes. Vis. Comput.
**1994**, 10, 353–355. [Google Scholar] [CrossRef] - Bhaniramka, P.; Wenger, R.; Crawfis, R. Isosurface construction in any dimension using convex hulls. IEEE Trans. Vis. Comput. Graph.
**2004**, 10, 130–141. [Google Scholar] [CrossRef] - Chojnacki, M.; Kisiel, A.; Florkowski, W.; Broniowski, W. THERMINATOR 2: THERMal heavy IoN generATOR 2. Comput. Phys. Commun.
**2012**, 183, 746–773. [Google Scholar] [CrossRef] [Green Version] - Bernhard, J.E. Bayesian Parameter Estimation for Relativistic Heavy-Ion Collisions. Ph.D. Thesis, Duke University, Durham, NC, USA, 2018. [Google Scholar]
- Shu, C.W. WENO methods. Scholarpedia
**2011**, 6, 9709. [Google Scholar] [CrossRef] - Thompson, K.W. The special relativistic shock tube. J. Fluid Mech.
**1986**, 171, 365–375. [Google Scholar] [CrossRef] - Sinyukov, Y.M.; Karpenko, I.A. Ellipsoidal flows in relativistic hydrodynamics of finite systems. Acta Phys. Hung. Ser. A Heavy Ion Phys.
**2006**, 25, 141–147. [Google Scholar] [CrossRef] [Green Version] - Chojnacki, M.; Florkowski, W.; Csörgö, T. Formation of Hubble-like flow in little bangs. Phys. Rev. C
**2005**, 71, 044902. [Google Scholar] [CrossRef] [Green Version] - Bazavov, A.; Bhattacharya, T.; DeTar, C.; Ding, H.-T.; Gottlieb, S.; Gupta, R.; Hegde, P.; Heller, U.M.; Karsch, F.; Laermann, E.; et al. Equation of state in ( 2+1 )-flavor QCD. Phys. Rev. D
**2014**, 90, 094503. [Google Scholar] [CrossRef] [Green Version] - Bożek, P. Effect of bulk viscosity on interferometry correlations in ultrarelativistic heavy-ion collisions. Phys. Rev. C
**2017**, 95, 054909. [Google Scholar] [CrossRef] [Green Version] - Bożek, P.; Broniowski, W.; Rybczynski, M.; Stefanek, G. GLISSANDO 3: GLauber Initial-State Simulation AND mOre..., ver. 3. Comput. Phys. Commun.
**2019**, 245, 106850. [Google Scholar] [CrossRef] [Green Version] - Bernhard, J. Frzout -Particlization Model (Cooper-Frye Sampler) for Relativistic Heavy-Ion Collisions. Available online: http://qcd.phy.duke.edu/frzout/index.html (accessed on 19 March 2020).

**Figure 4.**Evolution of energy density cross-section in XY plane, 30–40% most central events. (

**Left**) panel time of the initial stage 0.6 fm. (

**Right**) panel time of the final stage 9.6 fm.

**Figure 5.**Shapes of thefreeze-out hypersurface from the hydrodynamic simulations (Pb-Pb collisions at $\sqrt{{s}_{NN}}=2.76$ TeV, 0–2% most central). Red solid line result from own Hydro on GPU and the initial conditions were generated from the UrQMD, averaging over 100 events. model. Blue dashed line result based on Ref. [48] from a Hydro3p1 and initial conditions were generated from the Glissando program.

**Figure 6.**Shapes of the freeze-out hypersurface from the hydrodynamic simulations (Pb-Pb collisions at $\sqrt{{s}_{NN}}=2.76$ TeV, 0–2% most central). Red solid line represents a result from our hydrodynamic model on GPU using UrQMD model as an initial conditions generator. Left panel–averaging over 10 UrQMD events, right panel–single UrQMD event.

**Figure 7.**Distribution of the azimuth angle. Azimuth angle distribution as a result of the hydrodynamic simulation on the GPU for 2000 UrQMD events. Green dashed line is 0–2%, blue dashed line is 0–5%, and red solid line is 30–40% most central events.

**Figure 8.**Elliptic flow (${v}_{2}$) as a function transverse momentum ${p}_{T}$. Green dashed line is 0–2%, blue dashed line is 0–5%, and red solid line is 30–40% most central events (2000 events for each case).

Method | Time (s) (w/Copy Overhead) | Cells Intersections | Hypersurface Elements | % of Program Runtime (w/Copy Overhead) |
---|---|---|---|---|

Cornelius (single-threded) | 46 (46.2) | 6296765 | 6606727 | 89.1% (89.3%) |

Marching cubes (parallel) | 0.18 | 6296765 | 6669758 | 3.2% |

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**MDPI and ACS Style**

Słodkowski, M.; Setniewski, D.; Aszklar, P.; Porter-Sobieraj, J.
Modeling the Dynamics of Heavy-Ion Collisions with a Hydrodynamic Model Using a Graphics Processor. *Symmetry* **2021**, *13*, 507.
https://doi.org/10.3390/sym13030507

**AMA Style**

Słodkowski M, Setniewski D, Aszklar P, Porter-Sobieraj J.
Modeling the Dynamics of Heavy-Ion Collisions with a Hydrodynamic Model Using a Graphics Processor. *Symmetry*. 2021; 13(3):507.
https://doi.org/10.3390/sym13030507

**Chicago/Turabian Style**

Słodkowski, Marcin, Dominik Setniewski, Paweł Aszklar, and Joanna Porter-Sobieraj.
2021. "Modeling the Dynamics of Heavy-Ion Collisions with a Hydrodynamic Model Using a Graphics Processor" *Symmetry* 13, no. 3: 507.
https://doi.org/10.3390/sym13030507