1. 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,
「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  -----

2. Hydrodynamic Model
One of successful models for description of dynamics of QGP:
collisions
thermalization
hydro
hadronization
freezeout
observables
strong elliptic
hydrodynamic model

3. Viscosity in Hydrodynamics
Elliptic Flow
RHIC Au+Au GeV
Song et al, PRL106,192301(2011)
Reaction
plane x z y
0.08 < h/s < 0.24
Elliptic Flow

4. Ridge Structure Long correlation in longitudinal direction
1+1 d viscous hydrodynamics

5. 1+1 d relativistic viscous hydrodynamics
Fukuda

6. Perturvative calculation
Fukuda

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

8. Results
Fukuda

9. Viscosity in Hydrodynamics
Elliptic Flow
RHIC Au+Au GeV
Song et al, PRL106,192301(2011)
Reaction
plane x z y
0.08 < h/s < 0.24
Elliptic Flow

10. Higher Harmonics Higher harmonics and Ridge structure
Mach-Cone-Like structure, Ridge structure
Challenge to relativistic hydrodynamic model
Viscosity effect from initial en to final vn
Longitudinal structure (3+1) dimensional
Higher harmonics high accuracy calculations
State-of-the-art numerical algorithm
Shock-wave treatment
Less numerical dissipation

11. Hydrodynamic Model
One of successful models for description of dynamics of QGP:
collisions
thermalization
hydro
hadronization
freezeout
observables
higher harmonics
strong elliptic
particle yields:
PT distribution
fluctuating
initial conditions
hydrodynamic model
final state interactions:
hadron base
event generators
model
Viscosity, Shock wave

12. Current Status of Hydro
Ideal

13. 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

14. Numerical Scheme Lessons from wave equation Hydrodynamic 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
Wave equation: simple, analytical solution KT

15. Our Approach (COGNAC) Israel-Stewart Theory (ideal hydro)
Takamoto and Inutsuka, arXiv:
Israel-Stewart Theory
Akamatsu, Inutsuka, C.N., Takamoto,arXiv:
(ideal hydro)
1. dissipative fluid dynamics = advection + dissipation
exact solution for Riemann problem
Riemann solver: Godunov method
Contact discontinuity t
Rarefaction wave tt
Shock wave
Two shock approximation
Mignone, Plewa and Bodo, Astrophys. J. S160, 199 (2005)
Rarefaction wave shock wave
2. relaxation equation = advection + stiff equation

16. discontinuity surface
Riemann Problem
Discretization
Riemann problem
Energy distribution
shock wave:
discontinuity surface

17. discontinuity surface
Riemann Problem
Discretization
Riemann problem
Energy distribution
example
shock wave:
discontinuity surface
shock wave
Initial
Condition

18. discontinuity surface
Riemann Problem
Discretization
Riemann problem
Energy distribution
example
shock wave:
discontinuity surface
Initial
Condition

19. discontinuity surface
Riemann Problem
Discretization
Riemann problem
Energy distribution
example
shock wave:
discontinuity surface
Initial
Condition

20. discontinuity surface
Riemann Problem
Discretization
Riemann problem
Energy distribution
example
shock wave:
discontinuity surface
shock wave
rarefaction
wave
contact
discontinuity
shock wave

21. COGNAC COGite Numerical Analysis of heavy-ion Collisions
Takamoto and Inutsuka, arXiv:
Israel-Stewart Theory
Akamatsu, Inutsuka, C.N., Takamoto,arXiv:
(ideal hydro)
1. dissipative fluid dynamics = advection + dissipation
exact solution for Riemann problem
Riemann solver: Godunov method
Contact discontinuity t
Rarefaction wave tt
Shock wave
Two shock approximation
Mignone, Plewa and Bodo, Astrophys. J. S160, 199 (2005)
Rarefaction wave shock wave
2. relaxation equation = advection + stiff equation

22. Numerical Scheme Israel-Stewart Theory 1. Dissipative fluid equation
Takamoto and Inutsuka, arXiv:
1. Dissipative fluid equation
2. Relaxation equation +
advection
stiff equation
I: second order terms

23. Relaxation Equation Numerical scheme + stiff equation advection
Takamoto and Inutsuka, arXiv:
Numerical scheme +
stiff equation
advection
up wind method
during Dt
~constant
Piecewise exact solution
fast numerical scheme

24. Comparison
Shock Tube Test : Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)
Analytical solution
Numerical schemes
SHASTA, KT, NT
Our scheme
T=0.4 GeV
v=0
EoS: ideal gas
T=0.2 GeV
v=0 10
Nx=100, dx=0.1

25. Energy Density
t=4.0 fm dt=0.04, 100 steps
COGNAC
analytic

26. Velocity
t=4.0 fm dt=0.04, 100 steps
COGNAC
analytic

27. q
t=4.0 fm dt=0.04, 100 steps
COGNAC
COGNAC
analytic

28. Artificial and Physical Viscosities
Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)
Antidiffusion terms : artificial viscosity
stability

29. Numerical Dissipation
Sound wave propagation p
fm-4
If numerical dissipation does
not exist
Cs0:sound velocity
dp=0.1 fm-4
1000
After one cycle: t=l/cs0
Vs(x,t)=Vinit(x-cs0t)
With finite numerical dissipation
Vs(x,t)≠Vinit(x-cs0t) -2 2 fm
l=2 fm
periodic boundary condition
L1 norm
after one cycle

30. Convergence Speed
Space and time discretization
Second order accuracy

31. Numerical Dissipation
from fit of calculated data 1
1000

32. hnum vs Grid Size Numerical dissipation:
Deviation from linear analyses (Llin)
Ex. Heavy Ion Collisions
l ~ 10 fm
0.1<h/s<1
Fluctuating initial condition
T=500 MeV
l ~ 1 fm
Dx << 0.25 – 0.82 fm
Dx << 0.8 – 2.6 fm

33. Viscosity in Hydrodynamics
Elliptic Flow
RHIC Au+Au GeV
Song et al, PRL106,192301(2011)
physical viscosity = input of hydro
0.08 < h/s < 0.24

34. Viscosity in Hydrodynamics
Elliptic Flow
RHIC Au+Au GeV
Song et al, PRL106,192301(2011)
With finite numerical dissipation
0.08 < h/s < 0.24 ?
physical viscosity ≠ input of hydro
physical viscosity = input of hydro + numerical dissipation
Checking grid size dependence is important.

35. To Multi Dimension Operational split and directional split
Operational split (C, S)

36. To Multi Dimension Operational split and directional split
Operational split (C, S)
Li : operation in i direction 2d 3d

37. Blast Wave Problems Initial conditions Velocity: |v|=0.9
Pressure distribution
(0.2*vx, 0.2*vy)
fm-4 1

38. Blast Wave Problems

39. Higher Harmonics
Initial conditions
Gluaber model
smoothed
fluctuating

40. Higher Harmonics Initial conditions at mid rapidity Gluaber model
smoothed
fluctuating
t=10 fm
t=10 fm

41. Viscosity Effect 14 initial Pressure distribution Ideal t~5 fm t~10 fm7
Viscosity 7
0.9
0.25

42. Viscous Effect initial Pressure distribution Ideal t~5 fm t~10 fm20
initial
Pressure distribution
Ideal
t~5 fm
t~10 fm
t~15 fm
fm-4
0.25
1.2 9
Viscosity
0.3
1.2 9

43. 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