Study of higher harmonics based on

(3+1)-d relativistic viscous hydrodynamics

Department of Physics, Nagoya University

Chiho NONAKA

November 15, 2012@ATHIC2012, Pusan, Korea

In Collaboration with Yukinao AKAMATSU, Makoto TAKAMOTO

C. NONAKA

Time Evolution of Heavy Ion Collisions

thermalization hydro hadronization freezeoutcollisions

strong elliptic flow @RHIC particle yields:PT distribution

higher harmonics observables

modelhydrodynamic model final state interactions:

hadron base event generators

fluctuating initial conditions Viscosity, Shock wave

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Higher Harmonics• Higher harmonics and Ridge structure

Mach-Cone-Like structure, Ridge structure

State-of-the-art numerical algorithm •Shock-wave treatment •Less numerical viscosity

Challenge to relativistic hydrodynamic modelViscosity effect from initial en to final vn Longitudinal structure (3+1) dimensionalHigher harmonics high accuracy calculations

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Viscous Hydrodynamic Model• Relativistic viscous hydrodynamic equation

– First order in gradient: acausality – Second order in gradient: systematic treatment is not established• Israel-Stewart• Ottinger and Grmela• AdS/CFT• Grad’s 14-momentum expansion• Renomarization group

• Numerical scheme– Shock-wave capturing schemes

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Numerical Scheme • Lessons from wave equation– First order accuracy: large dissipation– Second order accuracy : numerical oscillation -> artificial viscosity, flux limiter

• Hydrodynamic equation– Shock-wave capturing schemes: Riemann problem • Godunov scheme: analytical solution of Riemann

problem, Our scheme • SHASTA: the first version of Flux Corrected Transport

algorithm, Song, Heinz, Chaudhuri • Kurganov-Tadmor (KT) scheme, McGill

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Current Status of HydroIdeal

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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, Nonaka, Takamoto, Inutsuka, in preparation

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Numerical Scheme • Israel-Stewart Theory Takamoto and Inutsuka, arXiv:1106.1732

1. Dissipative fluid equation

2. Relaxation equation

I: second order terms

+

advection stiff equation

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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)

T=0.4 GeVv=0

T=0.2 GeVv=0

0 10

Nx=100, dx=0.1

•Analytical solution

•Numerical schemes SHASTA, KT, NT Our scheme

EoS: ideal gas

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Energy Densityt=4.0 fm dt=0.04, 100 steps

analytic

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Velocity t=4.0 fm dt=0.04, 100 steps

analytic

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q t=4.0 fm dt=0.04, 100 steps

analytic

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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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Viscosity Effect

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EoS Dependence

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To Multi Dimension• Operational split and directional split

Operational split (C, S)

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To Multi Dimension• Operational split and directional split

Operational split (C, S)

Li : operation in i direction

2d

3d

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Higher Harmonics• Initial conditions– Gluaber model

smoothed fluctuating

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Higher Harmonics• Initial conditions at mid rapidity– Gluaber model

smoothed fluctuated

t=10 fm t=10 fm

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Time Evolution of vn

v2 is dominant. Smoothed IC Fluctuating IC

vn becomes finite.

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Time Evolution of Higher HarmonicsPetersen et al, Phys.Rev. C82 (2010) 041901

Ideal hydrodynamic calculationat mid rapidity en, vn: Sum up with entropy density weightEoS: ideal gas

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Viscosity Effect Pressure distribution

Viscosity

Ideal

initial

t~5 fm t~10 fm t~15 fm7 1 1

0.250.97

14fm-4

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Viscous Effectinitial Pressure distribution

Ideal t~5 fm t~10 fm t~15 fm

Viscosity

9 1.2 0.25

0.31.29

20

fm-4

fm-4

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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 – Fast numerical scheme

• Higher harmonics– Time evolution of en and vn

• Work in progress– Comparison with experimental data

Our algorithm

Akamatsu

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Backup

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Numerical MethodTakamoto and Inutsuka, arXiv:1106.1732

SHASTA, rHLLE, KT

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2D Blast Wave Check

Application to Heavy Ion collisionsAt QM2012!!

t=0 Pressure const.Velocity |v|=0.9 Velocity vectors（ t=0)

Shock wave

50 steps

100 steps500 steps

1000 steps

Numerical scheme, in preparationAkamatsu, Nonaka and Takamoto

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rHLLE vs SHASTASchneider et al. J. Comp.105(1993)92