Pseudorapidity distribution and decorrelation of anisotropic flow within the open-computing-language implementation CLVisc hydrodynamics

Studies of fluctuations and correlations of soft hadrons and hard and electromagnetic probes of the dense and strongly interacting medium require event-by-event hydrodynamic simulations of high-energy heavy-ion collisions that are computing intensive. We develop a $\left(3+1\right)$-dimensional viscous hydrodynamic model—CLVisc that is parallelized on a graphics processing unit (GPU) by using the open computing language (OpenCL) with 60 times performance increase for spacetime evolution and more than 120 times for the Cooper–Frye particlization relative to that without GPU parallelization. The model is validated with comparisons with different analytic solutions, other existing numerical solutions of hydrodynamics, and experimental data on hadron spectra in high-energy heavy-ion collisions. The pseudorapidity dependence of anisotropic flow $v_n\left(\eta \right)$ are then computed in CLVisc with initial conditions given by the a multiphase transport (ampt) model, with energy density fluctuations both in the transverse plane and along the longitudinal direction. Although the magnitude of $v_n\left(\eta \right)$ and the ratios between $v_2\left(\eta \right)$ and $v_3\left(\eta \right)$ are sensitive to the effective shear viscosity over entropy density ratio ${\eta }_v/s$, the shape of the $v_n\left(\eta \right)$ distributions in $\eta$ do not depend on the value of ${\eta }_v/s$. The decorrelation of $v_n$ along the pseudorapidity direction due to the twist and fluctuation of the event planes in the initial parton density distributions is also studied. The decorrelation observable $r_n({\eta }^a,{\eta }^b)$ between $v_n\{-{\eta }^a\}$ and $v_n\left\{{\eta }^a\right\}$ with the auxiliary reference window ${\eta }^b$ is found not to be sensitive to ${\eta }_v/s$ when there is no initial fluid velocity. For small ${\eta }_v/s$, the initial fluid velocity from mini-jet partons introduces sizable splitting of $r_n({\eta }^a,{\eta }^b)$ between the two reference rapidity windows ${\eta }^b\in [3,4]$ and ${\eta }^b\in [4.4,5.0]$, as has been observed in experiment. The implementation of CLVisc and guidelines on how to efficiently parallelize scientific programs on GPUs are also provided.

Figure 1

The spacetime coordinates of bad cells where $\max \left(\right|{\pi }^{\mu \nu }\left|\right)>T_0^{\tau \tau }$ from one testing event in $\sqrt{s_{NN}}=200\mathrm{GeV}\text{Au}+\text{Au}$ collisions at 0%–10% centrality. The bad cells in this event appear at $\tau \sim 10.62\mathrm{fm}$ with spatial coordinates far away from the freeze-out boundary.

Figure 2

Pressure as a function of energy density for five different equations of state. They are denoted as EOSI, lattice-wb2014, s95p-pce, EOSQ and pure gauge from top to bottom.

Figure 3

Pseudorapidity distributions for charged hadrons and identified particles ${\pi }^{+},K^{+}$ and proton from smooth particle spectra (black solid line) with integral resonance decay and Monte Carlo sampling (red dashed line) with forced resonance decay. The hydrodynamic evolution is given by CLVisc with optical Glauber initial condition at impact parameter $b=2.4\mathrm{fm}$, with initial time ${\tau }_0=0.4\mathrm{fm}$, the maximum energy density in most-central collisions ${\epsilon }_{\max }=55{\mathrm{GeV}/\mathrm{fm}}^3$ and lattice QCD EoS from the Wuppertal–Budapest 2014 computation.

Figure 4

The transverse momentum distribution for identified particles ${\pi }^{+},K^{+}$ and proton from smooth particle spectra (black solid line) with integral resonance decay and Monte Carlo sampling (red dashed line) with forced resonance decay. The hydrodynamic evolution is the same as in Fig. 3.

Figure 5

Comparison between CLVisc and Riemann solution for energy density evolution as a function of time.

Figure 6

Comparison between CLVisc and Riemann solution for fluid velocity evolution as a function of time.

Figure 7

Comparison between CLVisc and Bjorken solution for viscous hydrodynamics.

Figure 8

Time evolution of energy density distribution from CLVisc numerical results (solid) and Gubser analytical solution (dashed) for second-order viscous hydrodynamics.

Figure 9

Time evolution of transverse fluid velocity from CLVisc numerical results (solid) and Gubser analytical solution (dashed) for second-order viscous hydrodynamics.

Figure 10

Time evolution of ${\pi }^{xx}$ from CLVisc numerical results and Gubser analytical solution for second-order viscous hydrodynamics.

Figure 11

Time evolution of $-{\tau }^2{\pi }^{{\eta }_s{\eta }_s}$ from CLVisc numerical results and Gubser analytical solution for second-order viscous hydrodynamics.

Figure 12

Comparison between CLVisc (symbol points) and vish$2+1$ (lines) results for elliptic flow of direct ${\pi }^{+}$ in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with the optical Glauber initial condition at impact parameter $b=7\mathrm{fm}$ and with different values of shear-viscosity-to-entropy ratio. Results without the viscous correction $\delta f$ to the local phase-space distributions [Eq. (60)] are also shown.

Figure 13

Comparison between CLVisc (symbol points) and vish$2+1$ (lines) results for mean transverse fluid velocity $\langle v_r\rangle$ in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with the optical Glauber initial condition at impact parameter $b=7\mathrm{fm}$ and with different values of shear-viscosity-to-entropy-density ratio.

Figure 14

Comparison between CLVisc (symbols points) and vish$2+1$ (lines) results for momentum eccentricity in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with the optical Glauber initial condition at impact parameter $b=7\mathrm{fm}$ and with different values of shear-viscosity-to-entropy-density ratio.

Figure 15

Pseudorapidity distribution for charged hadrons in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with centrality range 0%–6%, 6%–15%, 15%–25%, and 25%–35%, from CLVisc with freeze-out temperature 100 MeV (solid lines) and 137 MeV (dashed lines) as compared with RHIC experimental data by the PHOBOS collaboration [116].

Figure 16

Invariant yield of ${\pi }^{+}$ in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with centrality range 0%–5%, 10%–15%, 20%–30%, and 30%–40%, from CLVisc with freeze-out temperature 100 MeV (solid lines) and 137 MeV (dashed lines) as compared with RHIC experimental data by PHENIX collaboration.

Figure 17

Pseudorapidity distribution for charged hadrons in $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\mathrm{TeV}$ with centrality range 0%–5%, 5%–10%, 10%–20%, and 20%–30%, from CLVisc (solid lines) and LHC experimental data by the ALICE collaboration [116].

Figure 18

$p_T$ spectra of charged pions for $\text{Pb}+\text{Pb}\sqrt{s_{NN}}=2.76\mathrm{TeV}$ collisions at centrality range 0%–5%, 5%–10%, 10%–20%, 20%–40%, 40%–60%, and 60%–80%, from CLVisc (solid lines) and LHC experimental data by the ALICE collaboration [117].

Figure 19

The centrality dependence of the anisotropic flows $v_2,v_3,v_4$, and $v_5$ from scalar-product method in $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\mathrm{TeV}$ with centrality ranges 0%–5%, 5%–10%, 10%–20%, and 20%–30%, from CLVisc (solid lines) and LHC experimental data (markers) by the ALICE collaboration [119].

Figure 20

The pseudorapidity dependence of elliptic flow and triangular flow, for $\text{Pb}+\text{Pb}\sqrt{s_{NN}}=2.76\mathrm{TeV}$ collisions with centrality range 0%–5%, 10%–20%, 20%–30%, 30%–40%, 40%–50%, and 50%–60%, from $\left(3+1\right)\text{D}$ viscous hydrodynamic simulations starting from ampt initial conditions without initial fluid velocity and evolve with ${\eta }_v/s=0.16$ as compared with LHC measurements from the ALICE collaboration [120].

Figure 21

(1a)–(1f) The decorrelation of elliptic flow and (2a)–(2f) triangular flow along the pseudorapidity direction, for $\text{Pb}+\text{Pb}\sqrt{s_{NN}}=2.76\mathrm{TeV}$ collisions with centrality range 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, and 40%–50%, from $\left(3+1\right)\text{D}$ viscous hydrodynamic simulations starting from ampt initial conditions without the initial fluid velocity $({\eta }_v/s=0$ for red lines and ${\eta }_v/s=0.16$ for blue circles and stars) as compared with LHC measurements at CMS (black squares). The label “ref1” denotes $3.0<{\eta }^b<4.0$, used in both CMS data and CLVisc calculation.

Figure 22

(1a)–(1f) The decorrelation of elliptic flow and (2a)–(2f) triangular flow along the pseudorapidity direction, for $\text{Pb}+\text{Pb}\sqrt{s_{NN}}=2.76\mathrm{TeV}$ collisions with centrality range 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, and 40%–50%, from $\left(3+1\right)\text{D}$ viscous hydrodynamic simulations starting from ampt initial conditions without the initial fluid velocity $({\eta }_v/s=0$ for red lines and ${\eta }_v/s=0.16$ for blue circles and stars) as compared with LHC measurements at CMS (black squares). The label “ref2” denotes $4.4<{\eta }^b<5.0$, used in both CMS data and CLVisc calculation.

Figure 23

The decorrelation of elliptic flow along the pseudorapidity direction, for $\text{Pb}+\text{Pb}\sqrt{s_{NN}}=2.76\mathrm{TeV}$ collisions with centrality range 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, and 40%–50%, from $\left(3+1\right)\text{D}$ viscous hydrodynamic simulations starting from ampt initial conditions with the initial fluid velocity $\mathrm{[}{\eta }_v/s=0$ for panels (1a)–(1f) and ${\eta }_v/s=0.16$ for panels (2a)–(2f)] as compared with LHC measurements at CMS (black squares). The label “ref1” denotes $3.0<{\eta }^b<4.0$, while “ref2” denotes $4.4<{\eta }^b<5.0$.

Figure 24

(1a)–(1f) The decorrelation of elliptic flow and (2a)–(2f) triangular flow ) along the pseudorapidity direction, for $\text{Au}+\text{Au}\sqrt{s_{NN}}=200\mathrm{GeV}$ collisions with centrality range 0%–5%, 5%–10%, 10%–20%, 20%–30%, 30%–40%, and 40%–50%, from $\left(3+1\right)\text{D}$ viscous hydrodynamic simulations starting from ampt initial conditions $({\eta }_v/s=0.16$ for red solid lines and ${\eta }_v/s=0$ for blue dashed lines) as compared with preliminary STAR measurements at RHIC (black squares). The reference rapidity window is $2.5<{\eta }^b<4.0$ and $y_{\text{beam}}=5.36$ for $\text{Au}+\text{Au}\sqrt{s_{NN}}=200\mathrm{GeV}$ collisions.

Figure 25

Cartoon diagram of the architecture of GPUs.

Figure 26

One strip of data stored in the shared memory for five-cell stencil in KT algorithm.

Figure 27

Parallel reduction used on GPU to compute the summation of particle spectra from millions of freeze-out hypersurface elements.