相対論的流体模型を軸にした重イオン衝突の理解

Kobayashi-Maskawa Institute Department of Physics, Nagoya University

Chiho NONAKA

June 23, 2013@Matsumoto「 RHIC-LHC 高エネルギー原子核反応の物理研究会」

----- QGP の物理研究会 信州合宿 -----

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Hydrodynamic Model• One of successful models for description of dynamics of

QGP: thermalization hydro hadronization freezeoutcollisions

strong elliptic flow @RHICobservables

hydrodynamic model

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Viscosity in HydrodynamicsSong et al, PRL106,192301(2011)Elliptic Flow

0.08 < h/s < 0.24Reaction plane

x

z

y

Elliptic Flow

RHIC Au+Au GeV

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Ridge Structure

Long correlation in longitudinal direction

1+1 d viscous hydrodynamics

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1+1 d relativistic viscous hydrodynamicsFukuda

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Perturvative calculationFukuda

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Perturbative Solution• F(1) の解：グリーン関数で構成される

Fukuda

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ResultsFukuda

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Viscosity in HydrodynamicsSong et al, PRL106,192301(2011)Elliptic Flow

0.08 < h/s < 0.24Reaction plane

x

z

y

Elliptic Flow

RHIC Au+Au GeV

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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 dissipation

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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Hydrodynamic Model• One of successful models for description of dynamics of

QGP: 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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Current Status of HydroIdeal

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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– Less numerical dissipation

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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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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 for Riemann problem

Rarefaction waveShock wave

Contact discontinuitytt

Rarefaction wave shock wave

Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665

t

(COGNAC)

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Riemann Problem• Discretization

Riemann problemEnergy distribution

shock wave: discontinuity surface

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Riemann Problem• Discretization

Riemann problemEnergy distribution

shock wave: discontinuity surface

Initial Condition

example

shock wave

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Riemann Problem• Discretization

Riemann problemEnergy distribution

shock wave: discontinuity surface

example

Initial Condition

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Riemann Problem• Discretization

Riemann problemEnergy distribution

shock wave: discontinuity surface

example

Initial Condition

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Riemann Problem• Discretization

Riemann problemEnergy distribution

shock wave: discontinuity surface

example

shock wave

shock wave

rarefactionwave

contact discontinuity

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COGNAC COGite Numerical Analysis of heavy-ion Collisions

• Israel-Stewart TheoryTakamoto 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 for Riemann problem

Rarefaction waveShock wave

Contact discontinuitytt

Rarefaction wave shock wave

Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665

t

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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 Density

analytic

t=4.0 fm dt=0.04, 100 steps

COGNAC

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Velocity

analytic

t=4.0 fm dt=0.04, 100 steps

COGNAC

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q

analytic

t=4.0 fm dt=0.04, 100 steps

COGNAC COGNAC

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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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Numerical Dissipation• Sound wave propagation

1000

0 2-2fm

fm-4

dp=0.1 fm-4

periodic boundary condition

Cs0:sound velocity

l=2 fm

After one cycle: t= /l cs0

Vs(x,t)=Vinit(x-cs0t)

If numerical dissipation does not exist

p

Vs(x,t)≠Vinit(x-cs0t)

With finite numerical dissipation

L1 norm

after one cycle

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Convergence Speed

Space and time discretizationSecond order accuracy

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Numerical Dissipation

1000 1

•numerical dissipation:

• from fit of calculated data

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hnum vs Grid Size

Numerical dissipation:Deviation from linear analyses (Llin)

Ex. Heavy Ion Collisions

l ~ 10 fm

0.1< /h s<1

T=500 MeV

Dx << 0.8 – 2.6 fm

Fluctuating initial condition

l ~ 1 fmDx << 0.25 – 0.82 fm

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Viscosity in HydrodynamicsSong et al, PRL106,192301(2011)Elliptic Flow

0.08 < h/s < 0.24physical viscosity = input of hydro

RHIC Au+Au GeV

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Viscosity in HydrodynamicsSong et al, PRL106,192301(2011)Elliptic Flow

0.08 < h/s < 0.24

RHIC Au+Au GeV

physical viscosity ≠ input of hydroWith finite numerical dissipation

physical viscosity = input of hydro + numerical dissipation

Checking grid size dependence is important.

?

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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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Blast Wave Problems• Initial conditions

Pressure distributionVelocity: |v|=0.9

(0.2*vx, 0.2*vy)

1

fm-4

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Blast Wave Problems

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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 fluctuating

t=10 fm t=10 fm

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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, COGNAC

– Viscosity effect– Shock wave capturing scheme: Godunov method

– Less numerical dissipation: crucial for viscosity analyses – Fast numerical scheme

• Numerical dissipation– How to evaluate numerical dissipation– Physical viscosity grid size

• Work in progress– Analyses of high energy heavy ion collisions– Realistic Initial Conditions + COGNAC + UrQMD

COGNAC

with Duke and Texas A&M