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 One of successful models for description of dynamics of QGP: collisions thermalization hydro hadronization freezeout observables strong elliptic flow @RHIC hydrodynamic model

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

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

1+1 d relativistic viscous hydrodynamics Fukuda

Perturvative calculation Fukuda

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

Results Fukuda

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

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

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

Current Status of Hydro Ideal

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

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

Our Approach (COGNAC) Israel-Stewart Theory (ideal hydro) Takamoto and Inutsuka, arXiv:1106.1732 Israel-Stewart Theory Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665 (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

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

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

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

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

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

COGNAC COGite Numerical Analysis of heavy-ion Collisions Takamoto and Inutsuka, arXiv:1106.1732 Israel-Stewart Theory Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665 (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

Numerical Scheme Israel-Stewart Theory 1. Dissipative fluid equation Takamoto and Inutsuka, arXiv:1106.1732 1. Dissipative fluid equation 2. Relaxation equation + advection stiff equation I: second order terms

Relaxation Equation Numerical scheme + stiff equation advection Takamoto and Inutsuka, arXiv:1106.1732 Numerical scheme + stiff equation advection up wind method during Dt ~constant Piecewise exact solution fast numerical scheme

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

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

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

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

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

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

Convergence Speed Space and time discretization Second order accuracy

Numerical Dissipation from fit of calculated data 1 1000

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

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

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.

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

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

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

Blast Wave Problems

Higher Harmonics Initial conditions Gluaber model smoothed fluctuating

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

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

Viscous Effect initial Pressure distribution Ideal t~5 fm t~10 fm 20 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

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