CLVisc: 3+1D Relativistic Hydrodynamic Framework

• CLVisc is a (3+1)D relativistic hydrodynamic framework that simulates heavy-ion collisions by evolving QCD matter from initial fluctuations to freeze-out using GPU-accelerated numerical schemes.
• It accommodates diverse initialization strategies and equations of state, including finite net baryon density and baryon diffusion, making it applicable to RHIC-BES and soft hadron phenomenology.
• The framework serves as the hydrodynamic backbone for complex observable calculations, ranging from flow decorrelations and heavy-flavor production to detailed spin polarization studies.

CLVisc is a $(3+1)$-dimensional relativistic hydrodynamic framework for heavy-ion collisions that serves as the spacetime evolution engine between fluctuating initial conditions and particlization. In the literature provided here, it appears as an event-by-event viscous hydrodynamic code, a GPU/OpenCL implementation of $(3+1)$D hydrodynamics, a finite-net-baryon-density extension with baryon diffusion, and the hydrodynamic backbone for calculations of soft hadrons, heavy flavor, directed flow, and hyperon spin polarization (Pang et al., 2018, Wu et al., 2021).

1. Historical development and scientific role

The framework is explicitly identified in the 2018 implementation paper as a $(3+1)$D viscous hydrodynamic model parallelized on GPUs using OpenCL, with “60 times performance increase for space-time evolution and more than 120 times for the Cooper-Frye particlization relative to that without GPU parallelization” (Pang et al., 2018). In that formulation, CLVisc is already positioned as an event-by-event tool for observables that depend on longitudinal fluctuations, such as $v_n(\eta)$ and longitudinal flow decorrelation.

A major later step is the finite-density extension to RHIC Beam Energy Scan conditions. That version adds net baryon number conservation, an Israel–Stewart-like equation for baryon diffusion, the NEOS-BQS equation of state, fluctuating $3$D Monte-Carlo Glauber initial conditions, and the SMASH afterburner, and is applied across $\sqrt{s_{NN}}=7.7$–$62.4$ GeV (Wu et al., 2021). This extension changes CLVisc from a baryon-free high-energy framework into a finite-$\mu_B$ baseline for BES phenomenology.

Across subsequent applications, CLVisc is used as the intermediate dynamical stage in multistage chains rather than as a standalone model. The cited papers couple it to AMPT for local spin polarization (Wu et al., 2020), to TRENTo for hydrodynamic $D$-meson production (Ding et al., 2021), to tilted optical Glauber initial conditions for directed flow (Jing et al., 2022), to SMASH or AMPT initial conditions for helicity polarization at BES energies (Yi et al., 2023), to TRENTo-3D or TRENTo3D for left-right splitting and polarization studies (Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026), and to KoMPoST-3D in a McDIPPER+KoMPoST-3D+CLVisc+SMASH chain for pre-equilibrium-to-hydrodynamics matching (Du et al., 8 Dec 2025). This corpus consistently presents CLVisc as the hydrodynamic response module that converts initial geometry, flow, and conserved-charge structure into freeze-out hypersurfaces and local gradients.

2. Hydrodynamic structure and evolved fields

In the general viscous formulation used in the CLVisc literature, hydrodynamic evolution is built on local conservation laws. Several papers write the system as

$$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$

with a dissipative decomposition of the energy-momentum tensor and current (Wu et al., 2021, Jiang et al., 8 Jun 2026). A finite-density version is written as

$(3+1)$0

while other CLVisc applications include bulk pressure explicitly through

$(3+1)$1

This is not a contradiction; it reflects paper-dependent choices about whether bulk viscosity or conserved charges are retained in a given application (Jing et al., 2022, Jiang et al., 8 Jun 2026).

The 2018 implementation paper states that CLVisc works in the Landau frame and uses Milne coordinates $(3+1)$2, with the Milne metric

$(3+1)$3

That paper also gives the projector orthogonal to the flow,

$(3+1)$4

and formulates the second-order viscous sector in an Israel–Stewart-type structure with shear-stress transversality and tracelessness (Pang et al., 2018).

For BES applications, CLVisc evolves baryon diffusion with an Israel–Stewart-like relaxation equation,

$(3+1)$5

together with a diffusion coefficient parameterized as

$(3+1)$6

The same finite-density paper gives

$(3+1)$7

and uses constant $(3+1)$8 and $(3+1)$9 across BES energies (Wu et al., 2021).

What CLVisc supplies to downstream analyses depends on the application, but the recurring output fields are the local temperature, flow velocity, conserved-charge information when present, dissipative tensors, and the freeze-out hypersurface. In the BES helicity-polarization study, the downstream calculation explicitly uses $(3+1)$0, $(3+1)$1, and $(3+1)$2 on the chemical freeze-out hypersurface, together with structures such as

$(3+1)$3

to decompose helicity polarization into thermal, shear, acceleration/temperature-gradient, and chemical components (Yi et al., 2023).

3. Numerical implementation, coordinates, and particlization

The explicit implementation paper makes clear that CLVisc is not only a hydrodynamic formalism but also a computational framework. It uses the second-order central Kurganov–Tadmor scheme for the convective part of the conservation laws and a two-step Runge–Kutta predictor-corrector update for the coupled viscous evolution (Pang et al., 2018). The same finite-density extension states that the hydrodynamic equations are solved with the Kurganov–Tadmor finite-volume algorithm and a second-order Runge–Kutta time integrator, again emphasizing low numerical viscosity and flux-based evolution (Wu et al., 2021).

GPU acceleration is a defining feature of the original implementation. The framework is parallelized with OpenCL, and the paper details acceleration of the hydrodynamic evolution, freeze-out hypersurface finding, and Cooper–Frye integration (Pang et al., 2018). A representative later performance example reports that, for a grid size

$(3+1)$4

a smooth-initial-condition $(3+1)$5–$(3+1)$6 Au+Au event at $(3+1)$7 GeV on an NVIDIA GeForce GTX 3080 takes about $(3+1)$8 s for the hydro event total, $(3+1)$9 s per time step, $v_n(\eta)$0 s per particle sample event, and $v_n(\eta)$1 s per SMASH afterburner event (Wu et al., 2021).

Particlization is consistently performed with Cooper–Frye. In the finite-density formulation this is written as

$v_n(\eta)$2

with the switching hypersurface found using the Cornelius routine (Wu et al., 2021). In the pre-equilibrium matching study, the CLVisc-to-SMASH handoff is specified at

$v_n(\eta)$3

again through Cooper–Frye with a shear viscous correction $v_n(\eta)$4 (Du et al., 8 Dec 2025). Other studies adopt constant-energy-density or constant-temperature criteria, for example

$v_n(\eta)$5

in several RHIC applications (Wu et al., 2021, Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026).

The implementation literature also shows that CLVisc is designed to support both smooth spectra integration and Monte Carlo sampling. The 2018 paper tabulates smooth spectra on

$v_n(\eta)$6

bins and describes Monte Carlo sampling that can interface with hadronic transport codes such as UrQMD, JAM, or SMASH (Pang et al., 2018). In practice, some studies use SMASH as an actual afterburner, whereas others include resonance decays but no hadronic rescattering (Wu et al., 2021, Jiang et al., 20 May 2025).

4. Initialization strategies, equations of state, and finite-density generalization

A central feature of CLVisc is its adaptability to multiple initial-condition models. The cited studies use optical Glauber (Wu et al., 2019, Jing et al., 2022), Monte-Carlo Glauber (Wu et al., 2021), AMPT (Pang et al., 2018, Wu et al., 2020), TRENTo (Ding et al., 2021), TRENTo-3D or TRENTo3D (Yi et al., 2024, Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026), SMASH initial conditions at BES energies (Yi et al., 2023), and a full pre-equilibrium energy-momentum tensor from KoMPoST-3D (Du et al., 8 Dec 2025). This variety is important: CLVisc is not tied to one initialization paradigm.

The same diversity appears in the equation of state. The implementation paper lists EOSI, lattice-wb2014, s95p-pce, EOSQ, and pure gauge as available choices, with s95p-pce used as the default in much of that paper’s phenomenology (Pang et al., 2018). The charm-hydrodynamics study uses a partial chemical equilibrium EoS with a chemical freeze-out temperature of $v_n(\eta)$7 MeV (Ding et al., 2021). The directed-flow paper uses the Wuppertal-Budapest 2014 lattice QCD EoS (Jing et al., 2022). Several recent top-RHIC and LHC applications use HotQCD-2014 or HotQCD-based equations of state (Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026). At BES energies, the finite-density extension uses NEOS-BQS, which is designed for matter with conserved baryon, electric charge, and strangeness content (Wu et al., 2021). The helicity-polarization study specifically selects NEOS-BQS because RHIC-BES matter is neither boost-invariant nor baryon-free (Yi et al., 2023).

The finite-density extension is the clearest statement of CLVisc as a conserved-charge hydrodynamic framework. It initializes both entropy density and net baryon density in $v_n(\eta)$8,

$v_n(\eta)$9

$3$0

with $3$1 fm and energy-dependent $3$2 chosen larger than the nuclear overlap time at low beam energy (Wu et al., 2021). By contrast, several top-energy studies explicitly set $3$3 and ignore baryon diffusion because midrapidity is approximately baryon-free in those settings (Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026).

Recent TRENTo3D applications show another direction of framework evolution: CLVisc can be initialized not only with scalar energy-density profiles but also with structured longitudinal momentum components. In the isobaric Zr+Zr polarization study, TRENTo3D is modified through

$3$4

so that an initial longitudinal flow gradient seeds vorticity before CLVisc evolution (Jiang et al., 8 Jun 2026). The KoMPoST-3D coupling goes further by passing a full out-of-equilibrium $3$5 into Landau-frame decomposition before CLVisc evolution (Du et al., 8 Dec 2025).

A practical implication, directly supported by the cited applications, is that there is no single universal CLVisc parameter set. Some papers give explicit values of $3$6, $3$7, $3$8, or $3$9, while others state only that parameters are adopted from earlier CLVisc studies or tuned to multiplicity and elliptic flow (Wu et al., 2020, Yi et al., 30 Aug 2025). This suggests that “CLVisc” names a hydrodynamic infrastructure rather than a unique phenomenological calibration.

5. Observable classes and phenomenological reach

CLVisc’s original benchmark domain is soft-hadron bulk phenomenology. The 2018 paper uses AMPT initial conditions with longitudinal fluctuations to compute $\sqrt{s_{NN}}=7.7$0 and the decorrelation observable

$\sqrt{s_{NN}}=7.7$1

finding that the magnitude of $\sqrt{s_{NN}}=7.7$2 depends on $\sqrt{s_{NN}}=7.7$3, but the shape of the $\sqrt{s_{NN}}=7.7$4 distributions does not (Pang et al., 2018). The BES finite-density extension broadens this to identified-particle spectra, mean transverse momenta, $\sqrt{s_{NN}}=7.7$5, $\sqrt{s_{NN}}=7.7$6, and $\sqrt{s_{NN}}=7.7$7 across $\sqrt{s_{NN}}=7.7$8–$\sqrt{s_{NN}}=7.7$9 GeV, providing what that paper describes as a benchmark for RHIC-BES soft-hadron data (Wu et al., 2021).

The framework is also used for observables that require explicitly longitudinal and rapidity-odd dynamics. In the directed-flow study, CLVisc evolves a tilted optical-Glauber initial condition and traces the chain from rapidity-dependent tilt to asymmetric pressure gradients, rapidity-odd $62.4$0, and final hadron $62.4$1 in Au+Au, Ru+Ru, and Zr+Zr collisions at $62.4$2 GeV (Jing et al., 2022). In the left-right elliptic-flow splitting analysis, TRENTo-3D+CLVisc is used to show that $62.4$3 is mainly controlled by $62.4$4, while $62.4$5 at midrapidity receives a substantial, even dominant, contribution from $62.4$6 (Jiang et al., 20 May 2025).

Heavy flavor appears in a more restrictive role. The $62.4$7-meson study uses event-by-event $62.4$8D viscous CLVisc as the hydrodynamic realization of an “extreme thermalization” scenario in which low-$62.4$9 charm quarks are assumed to comove with the fluid and hadronize statistically on the same freeze-out surface as light hadrons (Ding et al., 2021). In that setup, CLVisc reproduces low-$\mu_B$0 $\mu_B$1 yields and much of the measured $\mu_B$2, but overpredicts the spectrum at higher $\mu_B$3, supporting the paper’s conclusion that only the lowest-$\mu_B$4 charm sector is close to local thermal equilibrium.

A particularly extensive application domain is spin polarization. In the early local-polarization work, CLVisc provides $\mu_B$5, $\mu_B$6, and freeze-out hypersurfaces from which several candidate vorticity tensors are constructed and inserted into a polarization formula. Those studies show that the predicted local polarization depends strongly on the choice of spin chemical potential, with temperature-vorticity-based choices giving qualitatively coincidental agreement with RHIC measurements for some observables (Wu et al., 2019, Wu et al., 2020). At BES energies, the helicity-polarization study uses relativistic $\mu_B$7D CLVisc hydrodynamics with SMASH initial conditions and finds that the kinetic-vorticity contribution

$\mu_B$8

dominates over shear, acceleration/temperature-gradient, and chemical-potential-gradient terms at intermediate and low collision energies, especially at $\mu_B$9 GeV (Yi et al., 2023).

Recent spin studies push CLVisc into more differential observables. In Au+Au at $D$0 GeV and p+Pb at $D$1 TeV, CLVisc is used to compute

$D$2

under $D$3-equilibrium, $D$4-quark-equilibrium, and iso-thermal-equilibrium scenarios. The cited papers report that Au+Au data are described in the $D$5-quark and iso-thermal scenarios, whereas p+Pb remains a puzzle because the hydrodynamic thermal-vorticity and shear-induced contributions do not reproduce the observed signal (Yi et al., 2024, Yi et al., 30 Aug 2025). In isobaric $D$6 collisions, TRENTo3D + CLVisc with an initial longitudinal flow gradient provides a simultaneous description of STAR measurements of global polarization $D$7 and the azimuthal coefficients $D$8 and $D$9, while still overpredicting the high-$$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$0 modulation amplitude of $$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$1 (Jiang et al., 8 Jun 2026).

6. Limits of the framework and current extensions

The cited literature repeatedly distinguishes CLVisc as a hydrodynamic backbone from the observable-specific modules layered on top of it. This distinction is especially clear in the spin sector. The BES helicity-polarization paper states that spin polarization is computed on top of the hydrodynamic background rather than evolved as an independent hydrodynamic degree of freedom; no explicit spin-hydrodynamic backreaction is implemented, and no separate spin diffusion or axial-charge evolution is solved (Yi et al., 2023). This is important because it defines standard CLVisc polarization results as hydrodynamic baselines rather than complete spin-transport calculations.

The framework’s approximations are also strongly application dependent. Some studies use smooth optical-Glauber initial conditions and therefore address only rapidity-odd $$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$2, not rapidity-even components or event-by-event fluctuation effects (Jing et al., 2022). Several top-energy studies set $$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$3, omit baryon diffusion, and exclude hadronic rescattering after freeze-out (Jiang et al., 20 May 2025, Jiang et al., 8 Jun 2026). The finite-density BES baseline itself neglects bulk viscosity and has no explicit pre-equilibrium stage, with the authors noting that these omissions matter most strongly at the lowest beam energies (Wu et al., 2021).

Small systems expose another limitation. In p+Pb polarization studies, the authors explicitly warn that the lowest-multiplicity results are unreliable because of the limitations of hydrodynamic applicability, and they conclude that current hydrodynamic effects alone cannot explain the observed beam-direction spin polarization (Yi et al., 2024, Yi et al., 30 Aug 2025). This suggests that CLVisc can successfully generate realistic bulk backgrounds in small systems only up to the point where additional non-hydrodynamic or non-flow spin mechanisms become important.

Current extensions in the literature move in three directions. One is pre-equilibrium matching: the McDIPPER+KoMPoST-3D+CLVisc+SMASH study uses CLVisc as the downstream viscous stage after a non-boost-invariant kinetic-theory response calculation and finds that final observables become only weakly sensitive to the hydrodynamic switching time $$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$4 over a broad range (Du et al., 8 Dec 2025). A second direction is non-extensive hydrodynamics: NEX-CLVisc is developed “in the framework of CLVisc where the viscous corrections are turned off,” modifying initial conditions, the equation of state, and kinetic freeze-out through Tsallis-like non-extensive effects (Shi et al., 2024). A third direction is beyond-baseline spin physics: the recent local-polarization proceedings explicitly connect the present CLVisc-based calculations to interaction corrections from quantum kinetic theory and to spin hydrodynamics, while emphasizing that such corrections are not yet standard built-in components of CLVisc itself (Yi et al., 30 Aug 2025).

Taken together, these studies define CLVisc as a flexible $$\nabla_\mu T^{\mu\nu}=0, \qquad \nabla_\mu J^\mu=0,$$5D hydrodynamic infrastructure rather than a single fixed model. Its central identity is the evolution of realistic three-dimensional QCD matter, with viscous and, when needed, finite-density dynamics, from an externally specified initial state to a freeze-out hypersurface. Its scientific value lies in the fact that the same infrastructure can be coupled to very different initial-state models, equations of state, and observable modules; its limitations arise precisely where hydrodynamics, local equilibrium, or freeze-out-based post-processing become insufficient.