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A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions
Iu. Karpenko, P. Huovinen, M. Bleicher
TL;DR
Existing heavy-ion hydrodynamic codes often assume boost-invariant longitudinal expansion or zero net baryon density, limiting their applicability at Beam Energy Scan energies. This paper presents and tests a 3+1-dimensional viscous code that relaxes both assumptions, with results agreeing with analytical solutions and established benchmark codes.
Problem
Many heavy-ion hydrodynamic codes assume boost-invariant longitudinal expansion and/or zero net baryon density, which are poor approximations at Beam Energy Scan energies and forthcoming FAIR or NICA experiments.
Method
The code combines Godunov and relativistic HLLE treatment of the inviscid part with ideal-viscous splitting to solve Israel-Stewart viscous hydrodynamics in 3+1 dimensions.
Results
The numerical solutions agree with analytical inviscid and viscous solutions, while radial-expansion results are consistent with the VISH2+1 benchmark code.
Takeaways & Limitations
The code is intended for simulations of QCD matter expansion in relativistic heavy-ion collisions and was also checked against TECHQM test cases.
Takeaways & Limitations
The numerical-viscosity estimate does not guarantee the same viscosity in full 3+1-dimensional simulations, whose estimation in arbitrary geometry remains beyond the paper.
Abstract
from arXiv · showhide
We describe the details of 3+1 dimensional relativistic hydrodynamic code for the simulations of quark-gluon/hadron matter expansion in ultra-relativistic heavy ion collisions. The code solves the equations of relativistic viscous hydrodynamics in the Israel-Stewart framework. With the help of ideal-viscous splitting, we keep the ability to solve the equations of ideal hydrodynamics in the limit of zero viscosities using a Godunov-type algorithm. Milne coordinates are used to treat the predominant expansion in longitudinal (beam) direction effectively. The results are successfully tested against known analytical relativistic inviscid and viscous solutions, as well as against existing 2+1D relativistic viscous code.
1. Introduction
Relativistic hydrodynamics is important for modeling collective behavior in high-energy systems, including matter produced in ultrarelativistic heavy-ion collisions. The paper presents a 3+1-dimensional code addressing limitations of boost-invariant and zero-net-baryon-density assumptions relevant at lower collision energies.
- Motivation: Relativistic fluid dynamics describes high-energy phenomena ranging from astrophysical events to collective matter created in ultrarelativistic heavy-ion collisions.Ideal-fluid calculations have successfully described anisotropies in final particle distributions, motivating the near-perfect-fluid description of the matter.
- Research problem: Relativistic fluid-dynamics equations are difficult to solve numerically because analytic solutions exist only in highly idealized situations.The paper situates its code among existing numerical approaches for heavy-ion collisions.
- Research problem: Many existing heavy-ion codes assume boost-invariant longitudinal expansion and zero net baryon density throughout the system.These assumptions are not good approximations for Beam Energy Scan collisions at √sNN = 6.3–39 GeV or forthcoming FAIR and NICA experiments.
- Contribution: The authors develop a new code that relaxes both boost-invariant-expansion and zero-net-baryon-density assumptions.The paper presents test simulations of the resulting code.
- Method: The code uses the Godunov-type relativistic HLLE approximate Riemann solver, chosen for simplicity, reliability, and stability in ultrarelativistic heavy-ion simulations.The inviscid-fluid algorithm is used as the basis for studying nearly ideal fluids.
2. Equations
The code formulates relativistic viscous hydrodynamics using conservation laws, an externally supplied equation of state, and Israel-Stewart evolution equations. Milne-coordinate variables and source-term transformations are used to represent longitudinal expansion and solve the equations numerically.
- Formalism: The hydrodynamic equations follow from energy-momentum and charge conservation and are closed by an externally supplied equation of state p = p(ε, nc).The conserved charge current is indexed by c when multiple conserved charges are present.
- Formalism: The Landau frame defines flow velocity from the energy flow, and the viscous energy-momentum tensor includes energy density, pressure, shear stress, bulk pressure, and charge-diffusion currents.The paper explicitly identifies ε and p as rest-frame energy density and equilibrium pressure.
- Viscous dynamics: In the Israel-Stewart framework, shear stress and bulk pressure are independent dynamical variables with specified relaxation equations.The implementation neglects vorticity terms in the chosen equations of motion.
- Milne coordinates: Milne coordinates are used for the time-longitudinal plane, with longitudinal flow represented through rapidity variables and scaling Bjorken flow corresponding to uη = 0.These coordinates naturally describe expansion along the beam axis from a point-like source.
- Milne coordinates: Because Milne-coordinate source terms become dominant at small τ, the variables are redefined to separate factors of 1/τ before numerical evolution.The conserved variables, fluxes, viscous variables, and derivatives are rewritten in tilded form.
- Coordinate implementations: For shock-tube tests, the code uses Cartesian coordinates and solves the original hydrodynamic equations without Milne-coordinate transformations or geometrical source terms.This contrasts with the Milne-coordinate formulation used for most tests and heavy-ion simulations.
3. Numerical implementation
The numerical implementation combines finite-volume Godunov evolution, relativistic HLLE Riemann fluxes, second-order reconstruction and time stepping, and ideal-viscous splitting. Separate viscous evolution completes the Israel-Stewart update, with safeguards for unstable low-density or large-gradient regions.
- Ideal substep: The finite-volume method evolves cell-averaged conserved quantities through time-averaged boundary fluxes derived from conservation laws.Godunov evolution estimates these fluxes from exact or approximate Riemann problems at cell interfaces.
- Ideal substep: The timestep obeys the Godunov stability condition 2∆t ≤ ∆x after taking the maximal signal speed as c = 1.The same implementation also specifies signal velocities for the HLLE solver, including vacuum-facing cells.
- Viscous substep: Ideal-viscous splitting evolves ideal energy-momentum quantities with the Godunov method, propagates shear and bulk variables with Israel-Stewart equations, then applies viscous fluxes and sources.The splitting itself does not require viscous corrections to be small, although flux calculations assume the nearly perfect-fluid regime.
- Ideal substep: The ideal substep uses a relativistic HLLE approximate Riemann solver, which represents the discontinuity by one uniform intermediate state bounded by two waves.The solver is applied independently in the x, y, and η directions for three-dimensional evolution.
- Ideal substep: Second-order spatial accuracy is obtained with piecewise-linear MUSCL reconstruction and a minmod slope limiter that avoids introducing new extrema and numerical oscillations.A half-step predictor-corrector procedure provides second-order temporal accuracy.
- Viscous substep: Large gradients and high Lorentz factors can make Navier-Stokes viscous terms large and generate instabilities, so the code rescales dissipative variables when a prescribed bound is violated.Violations were found mainly in very low-density regions during expansion into vacuum, where viscous hydrodynamics may become inapplicable, while the dense core is unaffected in heavy-ion scenarios.
4. Test results
The code reproduces analytical ideal and viscous hydrodynamic solutions across shock-tube, vacuum-expansion, Gubser-flow, Bjorken, and heavy-ion collision tests. It also shows good directional consistency, benchmark agreement, energy conservation, and controlled numerical viscosity, with a stated limitation for full 3+1D viscosity estimates.
- Ideal hydrodynamics: The shock-tube solution becomes practically indistinguishable from the analytical result by timestep Nt = 999, while Nt = 25 shows substantial profile smearing.
- Ideal hydrodynamics: At Nt = 200, the 45-degree diagonal shock-tube propagation is consistent with the principal-direction solution.
- Ideal hydrodynamics: After Nt = 200 timesteps, the numerical energy-density and velocity profiles for expansion into vacuum approach the analytical solution.The rarefaction wave is spread over 100 hydrodynamic cells with λCFL = 0.5.
- Viscous hydrodynamics: The numerical solution accurately reproduces the challenging Gubser-flow evolution even after 10 fm/c.The tested effective transverse system size is about 1 fm, and the system expands and cools rapidly.
- Viscous hydrodynamics: The numerical temperature solution agrees with analytical inviscid and viscous Bjorken solutions, and the bulk-pressure solution reproduces both full and cut Israel-Stewart results.The bulk-pressure comparison uses analytical solutions with and without the 4Π/(3τ) term.
- Heavy-ion collision tests: Shear viscosity accelerates transverse radial flow, with results consistent with VISH2+1; total energy is conserved better than 3%.The viscous suppression of energy measures is attributed to collective-flow rearrangement and shear-stress contributions.
5. Conclusions
The paper presents and tests a 3+1-dimensional relativistic viscous hydrodynamic code based on Godunov-type methods and the HLLE approximation. The code solves Israel-Stewart viscous hydrodynamics, supports shock configurations, and targets QCD matter expansion in relativistic heavy-ion collisions.
- The code solves relativistic viscous hydrodynamics in the Israel-Stewart framework using ideal-viscous splitting.
- The Godunov method and relativistic HLLE approximation provide accurate treatment of shock wave configurations in the inviscid part.
- The authors test the code with shock tubes, Gubser flow, and two analytical viscous hydrodynamic solutions.
- The code’s primary application is simulating hydrodynamic expansion of QCD matter produced in relativistic heavy-ion collisions.
- The code is also checked against test cases developed by the TECHQM group.
Appendix A. Velocity finding
The velocity-finding procedure reconstructs rest-frame fluid quantities from conserved variables, accounting for shear and bulk viscous pressures. Symmetry reduces the recovery problem to a one-dimensional numerical equation for the velocity magnitude.
- The procedure recovers flow velocity, energy density, and conserved-charge densities in the fluid rest frame from conserved variables.
- For viscous evolution, the shear-stress tensor is subtracted from the total energy-momentum tensor before reconstructing fluid variables.
- The recovered variables are related through the fluid energy-momentum tensor and closed with an equation of state.
- Symmetry reduces velocity recovery to a one-dimensional equation for the absolute velocity, solved numerically.
- For a non-exotic equation of state, the velocity equation has exactly one root in [0,1) and is solved with Newton’s method.
- Bulk pressure is incorporated by replacing the equilibrium pressure p(ε, n_i) with p(ε, n_i) + Π.