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Study of higher harmonics based on (3+1)-d relativistic viscous hydrodynamics. Department of Physics, Nagoya University Chiho NONAKA. In Collaboration with Yukinao AKAMATSU, Makoto TAKAMOTO. November 15, 2012@ATHIC2012 , Pusan, Korea . Time Evolution of Heavy Ion Collisions. hydro. - PowerPoint PPT Presentation
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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
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 viscosity
Challenge to relativistic hydrodynamic modelViscosity effect from initial en to final vn Longitudinal structure (3+1) dimensionalHigher harmonics high accuracy calculations
C. NONAKA
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
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
Current Status of HydroIdeal
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
Rarefaction waveShock wave
Contact discontinuity
Rarefaction wave shock wave
Akamatsu, Nonaka, Takamoto, Inutsuka, in preparation
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 Densityt=4.0 fm dt=0.04, 100 steps
analytic
C. NONAKA
Velocity t=4.0 fm dt=0.04, 100 steps
analytic
C. NONAKA
q t=4.0 fm dt=0.04, 100 steps
analytic
C. NONAKA
Artificial and Physical ViscositiesMolnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)
Antidiffusion terms : artificial viscosity stability
C. NONAKA
Viscosity Effect
C. NONAKA
EoS Dependence
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
Higher Harmonics• Initial conditions– Gluaber model
smoothed fluctuating
C. NONAKA
Higher Harmonics• Initial conditions at mid rapidity– Gluaber model
smoothed fluctuated
t=10 fm t=10 fm
C. NONAKA
Time Evolution of vn
v2 is dominant. Smoothed IC Fluctuating IC
vn becomes finite.
C. NONAKA
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
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– 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
C. NONAKA
Backup
C. NONAKA
Numerical MethodTakamoto and Inutsuka, arXiv:1106.1732
SHASTA, rHLLE, KT
C. NONAKA
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
C. NONAKA
rHLLE vs SHASTASchneider et al. J. Comp.105(1993)92