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, YITP, Kyoto Hydrodynamic Model: Yukinao Akamatsu, Shu-ichiro Inutsuka, Makoto Takamoto Hybrid Model: Yukinao Akamatsu, Steffen Bass, Jonah Bernhard

C. NONAKA Numerical Algorithm Current understanding thermalization hydrohadronizationfreezeout collisions hydrodynamic model hadron based event generator fluctuating initial conditions

C. NONAKA Numerical Algorithm Current understanding thermalization hydrohadronizationfreezeout collisions hydrodynamic model hadron based event generator fluctuating initial conditions Initial geometry fluctuations MC-Glauber, MC-KLN, MC-rcBK, IP-Glasma… vnvn Hydrodynamic expansion

C. NONAKA Ollitrault

C. NONAKA Ollitrault Event-by-event calculation!

C. NONAKA Ollitrault Event-by-event calculation! small structure Shock-wave capturing scheme Stable, less numerical viscosity

C. NONAKA Numerical Algorithm Current understanding thermalization hydrohadronizationfreezeout collisions hydrodynamic model hadron based event generator fluctuating initial conditions Initial geometry fluctuations MC-Glauber, MC-KLN, MC-rcBK, IP-Glasma… vnvn Hydrodynamic expansion numerical method

C. NONAKA HYDRODYNAMIC MODEL Akamatsu, Inutsuka, CN, Takamoto: arXiv: 、 J. Comp. Phys. (2014) 34

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: 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

C. NONAKA Our Approach Israel-Stewart Theory Takamoto and Inutsuka, arXiv: 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 wave Shock wave Contact discontinuity Rarefaction wave shock wave Akamatsu, Inutsuka, CN, Takamoto, arXiv:

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

C. NONAKA Comparison Shock Tube Test : Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010) T L =0.4 GeV v=0 T R =0.2 GeV v= Nx=100, dx=0.1, dt=0.04 Analytical solution Numerical schemes SHASTA, KT, NT Our scheme EoS: ideal gas

C. NONAKA Shocktube problem Ideal case shockwave rarefaction

C. NONAKA L1 Norm Numerical dissipation: deviation from analytical solution N cell =100: dx=0.1 fm =10 fm For analysis of heavy ion collisions T L =0.4 GeV v=0 T R =0.2 GeV v=0 0 10

C. NONAKA Large  T difference T L =0.4 GeV, T R =0.172 GeV – SHASTA becomes unstable. – Our algorithm is stable. SHASTA: anti diffusion term, A ad – A ad = 1 : default value, unstable – A ad =0.99: stable, more numerical dissipation T L =0.4 GeV v=0 T R =0.172 GeV v= Nx=100, dx=0.1, dt=0.04 EoS: ideal gas

C. NONAKA L1 norm SHASTA with small A ad has large numerical dissipation Aad=1 Aad=0.99 T L =400, T R =200 T L =400, T R =172 =10 fm

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

C. NONAKA Large  T difference T L =0.4 GeV, T R =0.172 GeV – SHASTA becomes unstable. – Our algorithm is stable. SHASTA: anti diffusion term, A ad – A ad = 1 : default value – A ad =0.99: stable, more numerical dissipation Large fluctuation (ex initial conditions) – Our algorithm is stable even with small numerical dissipation. T L =0.4 GeV v=0 T R =0.172 GeV v= Nx=100, dx=0.1, dt=0.04 EoS: ideal gas

C. NONAKA HYBRID MODEL

C. NONAKA Our Hybrid Model thermalization hydrohadronizationfreezeout collisions hydrodynamic model MC-KLN Nara UrQMD CorneliusOscar sampler Freezeout hypersurface finder Huovinen, PetersenOhio group Fluctuating Initial conditionsHydrodynamic expansionFreezeout process From Hydro to particle Final state interactions Akamatsu, Inutsuka, CN, Takamoto, arXiv: 、 J. Comp. Phys. (2014) 34

C. NONAKA Our Hybrid Model thermalization hydrohadronizationfreezeout collisions hydrodynamic model MC-KLN Nara UrQMD CorneliusOscar sampler Freezeout hypersurface finder Huovinen, PetersenOhio group Fluctuating Initial conditionsHydrodynamic expansionFreezeout process From Hydro to particle Final state interactions Akamatsu, Inutsuka, CN, Takamoto, arXiv: 、 J. Comp. Phys. (2014) 34 Simulation setups: Free gluon EoS Hydro in 2D boost invariant simulation UrQMD with |y|<0.5

C. NONAKA Initial Pressure Distribution MC-KLN (centrality 15-20%) LHC RHIC Pressure (fm -4 ) X(fm) Y(fm)

C. NONAKA Time Evolution of Entropy Entropy of hydro (T>T SW =155MeV)

C. NONAKA Time Evolution of  n and v n Eccentricity & Flow anisotropy Shift the origin so that ε 1 =0

C. NONAKA Eccentricities vs higher harmonics LHC (200 events) nn vnvn

C. NONAKA Eccentricities vs higher harmonics RHIC (200 events) nn vnvn

C. NONAKA Hydro + UrQMD Transverse momentum spectrum LHCRHIC

C. NONAKA Effect of Hadronic Interaction Transverse momentum distribution LHCRHIC

C. NONAKA Higher harmonics from Hydro + UrQMD Effect of hadronic interaction

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 – 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