Viscous Hydrodynamics for Relativistic HeavyIon Collisions Riemann Solver

Viscous Hydrodynamics for Relativistic Heavy-Ion Collisions: Riemann Solver for Quark-Gluon Plasma Kobayashi Maskawa Institute Department of Physics, Nagoya University Chiho NONAKA Hydrodynamic Model: Yukinao Akamatsu, Shu-ichiro Inutsuka, Makoto Takamoto Hybrid Model: Yukinao Akamatsu, Steffen Bass, Jonah Bernhard December 5, 2013@NFQCD 2013, YITP, Kyoto

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

Numerical Algorithm • Current understanding collisions fluctuating initial conditions thermalization hydro hadronization hydrodynamic model Hydrodynamic expansion Initial geometry fluctuations MC-Glauber, MC-KLN, MC-rc. BK, IP-Glasma… C. NONAKA freezeout hadron based event generator vn

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 Ollitrault

Numerical Algorithm • Current understanding collisions fluctuating initial conditions thermalization hydro hadronization hydrodynamic model Hydrodynamic expansion Initial geometry fluctuations MC-Glauber, MC-KLN, numerical method MC-rc. BK, IP-Glasma… C. NONAKA freezeout hadron based event generator vn

Akamatsu, Inutsuka, CN, Takamoto: ar. Xiv: 1302. 1665、J. Comp. Phys. (2014) 34 HYDRODYNAMIC MODEL C. NONAKA

Viscous Hydrodynamic Model • Relativistic viscous hydrodynamic equation – First order in gradient: acausality – Second order in gradient: • Israel-Stewart, Ottinger and Grmela, Ad. S/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, Mc. Gill C. NONAKA

Our Approach • Israel-Stewart Theory Takamoto and Inutsuka, ar. Xiv: 1106. 1732 Akamatsu, Inutsuka, CN, Takamoto, ar. Xiv: 1302. 1665 (ideal hydro) 1. dissipative fluid dynamics = advection + dissipation exact solution Contact discontinuity Shock wave Rarefaction wave Riemann solver: Godunov method Two shock approximation Mignone, Plewa and Bodo, Astrophys. J. S 160, 199 (2005) Rarefaction wave 2. relaxation equation = advection + stiff equation C. NONAKA shock wave

Relaxation Equation Takamoto and Inutsuka, ar. Xiv: 1106. 1732 • Numerical scheme + stiff equation advection up wind method • during Dt ~constant Piecewise exact solution fast numerical scheme C. NONAKA

Comparison • Shock Tube Test : Molnar, Niemi, Rischke, Eur. Phys. J. C 65, 615(2010) • Analytical solution TL=0. 4 Ge. V v=0 Eo. S: ideal gas • Numerical schemes SHASTA, KT, NT Our scheme TR=0. 2 Ge. V v=0 10 0 Nx=100, dx=0. 1, dt=0. 04 C. NONAKA

Shocktube problem • Ideal case shockwave rarefaction C. NONAKA

L 1 Norm • Numerical dissipation: deviation from analytical solution TL=0. 4 Ge. V v=0 TR=0. 2 Ge. V v=0 10 0 For analysis of heavy ion collisions Ncell=100: dx=0. 1 fm l=10 fm C. NONAKA

Large DT difference • TL=0. 4 Ge. V, TR=0. 172 Ge. V – SHASTA becomes unstable. – Our algorithm is stable. TL=0. 4 Ge. V v=0 • SHASTA: anti diffusion term, Aad – Aad = 1 : default value, unstable 0 – Aad =0. 99: stable, more numerical dissipation C. NONAKA Eo. S: ideal gas TR=0. 172 Ge. V v=0 10 Nx=100, dx=0. 1, dt=0. 04

L 1 norm • SHASTA with small Aad has large numerical dissipation Aad=0. 99 Aad=1 TL=400, TR=200 TL=400, TR=172 l=10 fm C. NONAKA

Artificial and Physical Viscosities Molnar, Niemi, Rischke, Eur. Phys. J. C 65, 615(2010) Antidiffusion terms : artificial viscosity C. NONAKA stability

Large DT difference • TL=0. 4 Ge. V, TR=0. 172 Ge. V – SHASTA becomes unstable. – Our algorithm is stable. TL=0. 4 Ge. V v=0 • SHASTA: anti diffusion term, Aad – Aad = 1 : default value 0 – Aad =0. 99: stable, more numerical dissipation Eo. S: ideal gas TR=0. 172 Ge. V v=0 10 Nx=100, dx=0. 1, dt=0. 04 • Large fluctuation (ex initial conditions) – Our algorithm is stable even with small numerical dissipation. C. NONAKA

HYBRID MODEL C. NONAKA

Our Hybrid Model collisions thermalization Fluctuating Initial conditions hydro hadronization Hydrodynamic expansion Akamatsu, Inutsuka, CN, Takamoto, ar. Xiv: 1302. 1665、J. Comp. Phys. (2014) 34 MC-KLN hydrodynamic model Nara http: //www. aiu. ac. jp/~ynara/mckln/ C. NONAKA freezeout Freezeout process • From Hydro to particle • Final state interactions Cornelius Oscar sampler Freezeout hypersurface finder Huovinen, Petersen Ohio group Ur. QMD

Our Hybrid Model collisions thermalization Fluctuating Initial conditions hydro hadronization Hydrodynamic expansion Akamatsu, Inutsuka, CN, Takamoto, ar. Xiv: 1302. 1665、J. Comp. Phys. (2014) 34 MC-KLN hydrodynamic model Nara http: //www. aiu. ac. jp/~ynara/mckln/ C. NONAKA freezeout Freezeout process • From Hydro to particle • Final state interactions Cornelius Oscar sampler Freezeout hypersurface finder Huovinen, Petersen Simulation setups: • Free gluon Eo. S • Hydro in 2 D boost invariant simulation • Ur. QMD with |y|<0. 5 Ohio group Ur. QMD

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

Time Evolution of Entropy • Entropy of hydro (T>TSW=155 Me. V) C. NONAKA

Time Evolution of en and vn • Eccentricity & Flow anisotropy Shift the origin so that ε 1=0 C. NONAKA

Eccentricities vs higher harmonics • LHC (200 events) en C. NONAKA

Eccentricities vs higher harmonics • RHIC (200 events) en C. NONAKA

Hydro + Ur. QMD • Transverse momentum spectrum LHC C. NONAKA RHIC

Effect of Hadronic Interaction • Transverse momentum distribution LHC C. NONAKA RHIC

Higher harmonics from Hydro + Ur. QMD • Effect of hadronic interaction C. NONAKA

Summary Importance of numerical scheme • We develop a state-of-the-art numerical scheme – Viscosity effect – Shock wave capturing scheme: Godunov method Our algorithm – Less artificial diffusion: crucial for viscosity analyses – Stable for strong shock wave • Construction of a hybrid model – Fluctuating initial conditions + Hydrodynamic evolution + Ur. QMD • Higher Harmonics – Time evolution, hadron interaction C. NONAKA