Viscous Hydrodynamics for Relativistic Heavy-Ion Collisions:

Riemann Solver for Quark-Gluon Plasma

Kobayashi Maskawa Institute Department of Physics, Nagoya University

Chiho NONAKA

December 5, 2013@NFQCD 2013, YITP, Kyoto

Hydrodynamic Model: Yukinao Akamatsu, Shu-ichiro Inutsuka, Makoto Takamoto

Hybrid Model: Yukinao Akamatsu, Steffen Bass, Jonah Bernhard

C. NONAKA

Numerical Algorithm• Current understanding

thermalization hydro hadronization freezeoutcollisions

hydrodynamic model hadron based event generator

fluctuating initial conditions

C. NONAKA

Numerical Algorithm• Current understanding

thermalization hydro hadronization freezeoutcollisions

hydrodynamic model hadron based event generator

fluctuating initial conditions

Initial geometry fluctuations

MC-Glauber, MC-KLN,MC-rcBK, IP-Glasma…

vn

Hydrodynamic expansion

C. NONAKAOllitrault

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Event-by-event calculation!

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Event-by-event calculation!

small structureShock-wave capturing schemeStable, less numerical viscosity

C. NONAKA

Numerical Algorithm• Current understanding

thermalization hydro hadronization freezeoutcollisions

hydrodynamic model hadron based event generator

fluctuating initial conditions

Initial geometry fluctuations

MC-Glauber, MC-KLN,MC-rcBK, IP-Glasma…

vn

Hydrodynamic expansion

numerical method

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HYDRODYNAMIC MODEL

Akamatsu, Inutsuka, CN, Takamoto: arXiv:1302.1665、 J. Comp. Phys. (2014) 34

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Viscous Hydrodynamic Model• Relativistic viscous hydrodynamic equation

– First order in gradient: acausality – Second order in gradient:

• Israel-Stewart, Ottinger and Grmela, AdS/CFT, Grad’s 14-momentum expansion, Renomarization group

• Numerical scheme– Shock-wave capturing schemes: Riemann problem

• Godunov scheme: analytical solution of Riemann problem • SHASTA: the first version of Flux Corrected Transport

algorithm, Song, Heinz, Pang, Victor… • Kurganov-Tadmor (KT) scheme, McGill

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Our Approach • Israel-Stewart Theory

Takamoto and Inutsuka, arXiv:1106.1732

1. dissipative fluid dynamics = advection + dissipation

2. relaxation equation = advection + stiff equation

Riemann solver: Godunov method

(ideal hydro)

Mignone, Plewa and Bodo, Astrophys. J. S160, 199 (2005)

Two shock approximation

exact solution

Rarefaction waveShock wave

Contact discontinuity

Rarefaction wave shock wave

Akamatsu, Inutsuka, CN, Takamoto, arXiv:1302.1665

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Relaxation Equation• Numerical scheme

+

advection stiff equation

up wind method

Piecewise exact solution

~constant• during Dt

Takamoto and Inutsuka, arXiv:1106.1732

fast numerical scheme

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Comparison • Shock Tube Test : Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)

TL=0.4 GeVv=0

TR=0.2 GeVv=0

0 10

Nx=100, dx=0.1, dt=0.04

•Analytical solution

•Numerical schemes SHASTA, KT, NT Our scheme

EoS: ideal gas

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Shocktube problem• Ideal case

shockwave

rarefaction

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L1 Norm• Numerical dissipation: deviation from analytical solution

Ncell=100: dx=0.1 fm

l=10 fm

For analysis of heavy ion collisions

TL=0.4 GeVv=0

TR=0.2 GeVv=0

0 10

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Large DT difference • TL=0.4 GeV, TR=0.172 GeV– SHASTA becomes unstable. – Our algorithm is stable.

• SHASTA: anti diffusion term, Aad

– Aad = 1 : default value, unstable

– Aad =0.99: stable,

more numerical dissipation

TL=0.4 GeVv=0

TR=0.172 GeVv=0

0 10Nx=100, dx=0.1, dt=0.04

EoS: ideal gas

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L1 norm• SHASTA with small Aad has large numerical dissipation

Aad=1

Aad=0.99

TL=400, TR=200 TL=400, TR=172

l=10 fm

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Artificial and Physical ViscositiesMolnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)

Antidiffusion terms : artificial viscosity stability

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Large DT difference • TL=0.4 GeV, TR=0.172 GeV– SHASTA becomes unstable. – Our algorithm is stable.

• SHASTA: anti diffusion term, Aad

– Aad = 1 : default value

– Aad =0.99: stable,

more numerical dissipation

• Large fluctuation (ex initial conditions)– Our algorithm is stable even with small numerical dissipation.

TL=0.4 GeVv=0

TR=0.172 GeVv=0

0 10Nx=100, dx=0.1, dt=0.04

EoS: ideal gas

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HYBRID MODEL

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Our Hybrid Modelthermalization hydro hadronization freezeoutcollisions

hydrodynamic modelMC-KLN Nara

http://www.aiu.ac.jp/~ynara/mckln/

UrQMD

Cornelius Oscar samplerFreezeout hypersurface finder

Huovinen, Petersen Ohio group

Fluctuating Initial conditions Hydrodynamic expansion Freezeout process•From Hydro to particle•Final state interactions Akamatsu, Inutsuka, CN, Takamoto,

arXiv:1302.1665、 J. Comp. Phys. (2014) 34

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Our Hybrid Modelthermalization hydro hadronization freezeoutcollisions

hydrodynamic modelMC-KLN Nara

http://www.aiu.ac.jp/~ynara/mckln/

UrQMD

Cornelius Oscar samplerFreezeout hypersurface finder

Huovinen, Petersen Ohio group

Fluctuating Initial conditions Hydrodynamic expansion Freezeout process•From Hydro to particle•Final state interactions Akamatsu, Inutsuka, CN, Takamoto,

arXiv:1302.1665、 J. Comp. Phys. (2014) 34

Simulation setups:• Free gluon EoS• Hydro in 2D boost invariant simulation• UrQMD with |y|<0.5

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Initial Pressure Distribution• MC-KLN (centrality 15-20%)

LHC RHIC

Pressure (fm-4)

X(fm) X(fm)

Y(fm) Y(fm)

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Time Evolution of Entropy• Entropy of hydro (T>TSW=155MeV)

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Time Evolution of en and vn • Eccentricity & Flow anisotropy

Shift the origin so that ε1=0

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Eccentricities vs higher harmonics• LHC (200 events)

en vn

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Eccentricities vs higher harmonics• RHIC (200 events)

en vn

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Hydro + UrQMD• Transverse momentum spectrum

LHC RHIC

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Effect of Hadronic Interaction• Transverse momentum distribution

LHC RHIC

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Higher harmonics from Hydro + UrQMD• Effect of hadronic interaction

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Summary

• We develop a state-of-the-art numerical scheme– Viscosity effect– Shock wave capturing scheme: Godunov method

– Less artificial diffusion: crucial for viscosity analyses – Stable for strong shock wave

• Construction of a hybrid model– Fluctuating initial conditions + Hydrodynamic evolution +

UrQMD

• Higher Harmonics– Time evolution, hadron interaction

Our algorithm

Importance of numerical scheme