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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
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Numerical Algorithm• Current understanding
thermalization hydro hadronization freezeoutcollisions
hydrodynamic model hadron based event generator
fluctuating initial conditions
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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
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C. NONAKAOllitrault
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C. NONAKAOllitrault
Event-by-event calculation!
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Event-by-event calculation!
small structureShock-wave capturing schemeStable, less numerical viscosity
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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