Status of viscous hydrodynamic code development

Transcription

1 Status of viscous hydrodynamic code development Yuriy KARPENKO Transport group meeting, Jan 17, 2013 Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

2 Introduction: heavy ion collision in pictures1 Initial state, hard scatterings Thermalization Hydrodynamic expansion I I I I Quark-gluon plasma Phase transition Hadron Gas Chemical freeze-out Kinetic stage Typical size 10 fm m Typical lifetime 10 fm/c s (kinetic) freeze-out 1 taken from event generator Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

3 Relativistic viscous hydrodynamics { µ T µν = 0 µ N µ = 0 T µν = εu µ u ν (p + Π) µν + q µ u ν + q ν u µ + π µν N µ = n u µ + v µ +equation of state (EoS) p = p(ε,n) where µν = g µν u µ u ν is the projector orthogonal to u µ, (in the fluid rest frame where u µ = (1,0,0,0) µν = diag(0, 1, 1, 1)) q µ is the heat flux π µν and Π are the shear stress tensor and bulk pressure v µ is a (baryon, etc) charge diffusion q µ = π µν = Π = v µ = 0 ideal fluid. Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

4 Definition of u µ Since in general case there is no common direction for energy-momentum flow and particle flow, 1. Landau definition (flow of energy): u µ L = T ν µ ul ν ul αt α β T βγ u γ L q µ = 0 2. Eckart definition (flow of conserved charge) u µ E = N µ Nν N ν v µ = 0! we adopt Landau definition (Eckart frame is undefined when n = 0) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

5 Relativistic Navier-Stokes from the second law of thermodynamics In ideal hydrodynamics: { µ T µν = 0 µ N µ = 0 µ S µ = µ (su µ ) = 0 In viscous hydrodynamics: µ S µ > 0 T µ S µ =... = u ν µ δt µν =... = π µν < µλ λ u ν > Π µ u µ Then, µ S µ > 0 if { π µν = 2η < µλ λ u ν > Π = ζ µ u µ These constitutive equations correspond to relativistic generalization of Navier-Stokes equations. Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

6 Relativistic Navier-Stokes Causality problems: For small perturbations around uniform flow: t δu y Diffusion speed for wavemode k: η ε + p 2 x δu y = 0 v T (k) = 2k η ε + p, k 1 Such paradox is a consequence of an insufficient description of the thermodynamical state in nonequilibrium (Müller, 1968) Acausality instabilities (Hiscock, Lindblom 1985) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

7 Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

8 Second-order theory: Israel-Stewart formalism Entropy current S µ = su µ = s eq u µ (β 0 Π 2 + β 2 π αβ π αβ ) uµ 2T Then, the requirement µ S µ > 0 leads to < Dπ µν >= π µν π µν NS 1 ( τ π 2 π µν λ u λ + D ln β 0 T DΠ = Π Π NS τ Π 1 ( 2 Π λ u λ + D ln β ) 2 T where D u µ µ, τ π = 2ηβ 2, τ Π = ζ β 0, and Navier-Stokes values for the viscous components: π µν NS = 2ησ µν = η( µλ λ u ν + νλ λ u µ ) 2 3 η µν λ u λ Π NS = ζ λ u λ The solutions are stable, provided that τ π,τ Π are big enough (for example, τ Π > 3 ζ 2 st from stability condition around hydrostatic state ) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21 )

9 The scheme Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

10 Coordinate transformations Milne coordinates The coordinate system is defined as follows: 0) τ = t 2 z 2 1) x = x 2) y = y 3) η = 1 2 ln t + z t z g µν = diag(1, 1, 1, 1/τ 2 ) Nonzero Christoffel symbols are: Γ η τη = Γ η ητ = 1/τ, Γ τ ηη = τ T µν = (ε + p)u µ u ν p g µν, where u µ = {cosh(η f η)coshη T,sinhη T cosφ,sinhη T sinφ, 1 τ sinh(η f η)coshη T } (cf. u i Cart = {cosh(η f )coshη T,sinhη T cosφ,sinhη T sinφ,sinh(η f )coshη T }) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

11 EM conservation equations are ;ν T µν = 0 or µ = 0 : ν T τν + τt ηη + 1 τ T ττ = 0 µ = 1 : ν T xν + 1 τ T xτ = 0 µ = 2 : ν T yν + 1 τ T yτ = 0 µ = 3 : ν T ην + 3 τ T ητ = 0 Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

12 Additional transformations: EM conservation equations are ;ν T µν = 0 or µ = 0 : ν T τν + τt ηη + 1 τ T ττ = 0 µ = 1 : ν T xν + 1 τ T xτ = 0 µ = 2 : ν T yν + 1 τ T yτ = 0 µ = 3 : ν T ην + 3 τ T ητ = 0 T µη T µη /τ, µ η, T ηη T ηη /τ 2 ν (τt τν ) + 1 τ (τt ηη ) = 0 ν (τt xν ) = 0 ν (τt yν ) = 0 ν (τt ην ) + 1 τ τt ητ = 0 Conservative variables are Q µ = τ T τµ Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

13 The exact expressions for evolutionary equations for viscous corrections: where the extra source-terms are: γ ( t + v i i )π µν = π µν π µν NS τ π + I π (1) γ ( t + v i i )Π = Π Π NS τ Π + I Π (2) extra IS terms {}}{ <Dπ µν >=... {}}{ 4 I π = 3 π µν ;γ u γ [u ν π µβ + u µ π νβ ]u λ ;λ u β +I π,geom (π) (3) I Π = 4 3 Π ;γu γ + I Π,geom (Π) (4) and ;µ u ν = µ u ν + Γ ν µλ uλ is covariant derivative. Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

14 Closer to numerics: µ (T µν id + δt µν ) = S ν, S=geometrical source terms Finite-volume realization: τ (Tid τi + δt τi ) + j (T ji ) + }{{}}{{} j (δt ji ) = Sid ν }{{}} + {{ δsν } Q i id.flux visc.flux source terms 1 τ (Qn+1 id +δq n+1 Q n id δqn )+ 1 x ( F n+1/2 id + δf n+1/2 ) = S n+1/2 id +δs n+1/2 2 Makoto Takamoto, Shu-ichiro Inutsuka, J.Comput.Phys. 230 (2011), 7002 Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

15 Closer to numerics: µ (T µν id + δt µν ) = S ν, S=geometrical source terms Finite-volume realization: τ (Tid τi + δt τi ) + j (T ji ) + }{{}}{{} j (δt ji ) = Sid ν }{{}} + {{ δsν } Q i id.flux visc.flux source terms 1 τ (Qn+1 id +δq n+1 Qid n δqn )+ 1 x now, a small trick: 1 τ (Qn+1 id +δq n+1 Q n+1 id + Q n+1 id } {{ } =0 Then,split the equation into two parts 2 : ( F n+1/2 id + δf n+1/2 ) = S n+1/2 id +δs n+1/2 Q n id δqn )+ 1 x ( F id + δf) = S id +δs 1 t (Q n+1 id Qid n ) + 1 x F id = S id (5) 1 t (Qn+1 id + δq n+1 Q n+1 id δq n ) + 1 δf = δs x (6) 2 Makoto Takamoto, Shu-ichiro Inutsuka, J.Comput.Phys. 230 (2011), 7002 Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

16 The solution then proceeds in two stages: 1a) Q n+1 id is obtained by evolving only the ideal part (5) of energy-momentum tensor over the full timestep t, which is done accurately using Godunov method: rhlle flux + MUSCL + predictor-corrector schemes τu ν u (n+1),ν u n,ν τ x u ν u (n+1),ν k+1 u (n+1),ν k 1 2 x 1b) Advection-type evolutionary equations for π µν,π are solved with 1st order upwind scheme. Velocity gradients for Navier-Stokes values of π µν,π etc. are taken from above. 2) Q n+1 id,i + δq n+1 i = Q n+1 full is obtained from Eq. (6) evolving over the full timestep t with viscous fluxes/sources ONLY. *The initial condition for stage 2 is Q ini = Q n+1 id + δq n, the first term obtained from the solution of the stage 1. Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

17 test #0: 0+1D comparison with known analytical solution with viscosity in Navier-Stokes limit Energy conservation: ν T τν + τt ηη + 1 τ T ττ = 0 0+1D, u µ = 1,0,0,0, T µν id = diag(ε,p,p,p/τ2 ), the only nonzero π µν is τ 2 π ηη = 4 η 3 τ ε τ + ε + p + τ2 π ηη τ = 0 Assuming the EoS for relativistic maseless gas, ε = αt 4, s = 4 3 T ε, the solution is: ( τ0 ) [ 1/3 T (τ) = T (τ 0 ) + 2η ( ( τ0 ) )] 2/3 1 τ 3sτ 0 τ The same solution exists for bulk viscosity Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

18 setup for test calculations 3 #1 Glauber IC for energy density τ 0 = 0.6 fm/c, longitudinal boost-invariance p = ε/3 EoS Navier-Stokes values for initial π µν (nonzero due to Bjorken longitudinal flow) η/s = 1/(4π), ζ /s = 0 compared to ideal case 3 to compare with H. Song, PhD thesis, arxiv: Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

19 test #1: cooling rates for ideal/visc τ = 0.6 fm/c Graph τ = 4 fm/c ideal η/s=1/4π [GeV/fm^3] ideal η/s=1/4π τ = 1 fm/c [fm] r T τ = 7 fm/c [GeV/fm^3] ideal η/s=1/4π [GeV/fm^3] 0.25 ideal η/s=1/4π [fm] r T [fm] r T Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

20 test #1: additional transverse push & anizotropy suppression 0.14 < P >, with T only ideal µν 0.12 Central and noncentral collisions: Glauber IC, b = 7 fm [c]> <v T radial flow <v_t> ideal η/s=1/4π τ [fm/c] τ [fm/c] 0.09 < v - v > x y τ [fm/c] Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

21 test #2 Comparison with vshasta by E. Molnar 3D Gaussian for ε, τ 0 = 1 fm/c EoS p = ε/3 π µν ini = 0,Π ini = 0 no vacuum cells Energy density/velocity proifles agree π µν plotted at: 1.4 (left), 2.2 (top right), 3.8 fm/c (bottom right) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

22 Test #3: Fluctuating ICs: (typical EPOS event) Ideal hydro η/s = 3/(4π) Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21

23 Energy conservation checks has been made as well (trivial for η/s = 0 in Cartesian coordinates, nontrivial otherwise) Some speed and memory usage optimization, cleanup of the code, improving the structure etc. needs to be done Stay tuned! Thanks for your attention! Yuriy Karpenko (FIAS/BITP) Status of viscous hydro code Transport group meeting, Jan 17, / 21