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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 -----. Hydrodynamic Model. hydro. hadronization. freezeout. collisions. thermalization. - PowerPoint PPT Presentation
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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 -----
C. NONAKA
Hydrodynamic Model• One of successful models for description of dynamics of
QGP: thermalization hydro hadronization freezeoutcollisions
strong elliptic flow @RHICobservables
hydrodynamic model
C. NONAKA
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
C. NONAKA
Ridge Structure
Long correlation in longitudinal direction
1+1 d viscous hydrodynamics
C. NONAKA
1+1 d relativistic viscous hydrodynamicsFukuda
C. NONAKA
Perturvative calculationFukuda
C. NONAKA
Perturbative Solution• F(1) の解:グリーン関数で構成される Fukuda
C. NONAKA
ResultsFukuda
C. NONAKA
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
C. NONAKA
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
C. NONAKA
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
C. NONAKA
Current Status of HydroIdeal
C. NONAKA
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
C. NONAKA
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
C. NONAKA
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)
C. NONAKA
Riemann Problem• Discretization
Riemann problemEnergy distribution
shock wave: discontinuity surface
C. NONAKA
Riemann Problem• Discretization
Riemann problemEnergy distribution
shock wave: discontinuity surface
Initial Condition
example
shock wave
C. NONAKA
Riemann Problem• Discretization
Riemann problemEnergy distribution
shock wave: discontinuity surface
example
Initial Condition
C. NONAKA
Riemann Problem• Discretization
Riemann problemEnergy distribution
shock wave: discontinuity surface
example
Initial Condition
C. NONAKA
Riemann Problem• Discretization
Riemann problemEnergy distribution
shock wave: discontinuity surface
example
shock wave
shock wave
rarefactionwave
contact discontinuity
C. NONAKA
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
C. NONAKA
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
C. NONAKA
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
C. NONAKA
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
C. NONAKA
Energy Density
analytic
t=4.0 fm dt=0.04, 100 steps
COGNAC
C. NONAKA
Velocity
analytic
t=4.0 fm dt=0.04, 100 steps
COGNAC
C. NONAKA
q
analytic
t=4.0 fm dt=0.04, 100 steps
COGNAC COGNAC
C. NONAKA
Artificial and Physical ViscositiesMolnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)
Antidiffusion terms : artificial viscosity stability
C. NONAKA
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
C. NONAKA
Convergence Speed
Space and time discretizationSecond order accuracy
C. NONAKA
Numerical Dissipation
1000 1
•numerical dissipation:
• from fit of calculated data
C. NONAKA
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 conditionl ~ 1 fm
Dx << 0.25 – 0.82 fm
C. NONAKA
Viscosity in HydrodynamicsSong et al, PRL106,192301(2011)Elliptic Flow
0.08 < h/s < 0.24physical viscosity = input of hydro
RHIC Au+Au GeV
C. NONAKA
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.
?
C. NONAKA
To Multi Dimension• Operational split and directional split
Operational split (C, S)
C. NONAKA
To Multi Dimension• Operational split and directional split
Operational split (C, S)
Li : operation in i direction
2d
3d
C. NONAKA
Blast Wave Problems• Initial conditions
Pressure distributionVelocity: |v|=0.9
(0.2*vx, 0.2*vy)
1
fm-4
C. NONAKA
Blast Wave Problems
C. NONAKA
Higher Harmonics• Initial conditions– Gluaber model
smoothed fluctuating
C. NONAKA
Higher Harmonics• Initial conditions at mid rapidity– Gluaber model
smoothed fluctuating
t=10 fm t=10 fm
C. NONAKA
Viscosity Effect Pressure distribution
Viscosity
Ideal
initial
t~5 fm t~10 fm t~15 fm7 1 1
0.250.97
14fm-4
C. NONAKA
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
C. NONAKA
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