A new scheme of causal viscous hydrodynamics for relativistic heavy-ion collisions: A Riemann solver for quark-gluon plasma

In this article, we present a state-of-the-art algorithm for solving the relativistic viscous hydrodynamics equation with the QCD equation of state. The numerical method is based on the second-order Godunov method and has less numerical dissipation, which is crucial in describing of quark-gluon plasma in high-energy heavy-ion collisions. We apply the algorithm to several numerical test problems such as sound wave propagation, shock tube and blast wave problems. In sound wave propagation, the intrinsic numerical viscosity is measured and its explicit expression is shown, which is the second-order of spatial resolution both in the presence and absence of physical viscosity. The expression of the numerical viscosity can be used to determine the maximum cell size in order to accurately measure the effect of physical viscosity in the numerical simulation.

[1]  I. Karpenko,et al.  Kaon and pion femtoscopy at the highest energies available at the BNL Relativistic Heavy Ion Collider (RHIC) in a hydrokinetic model , 2010, 1004.1565.

[2]  T. Kodama,et al.  Topics on Hydrodynamic Model of Nucleus-Nucleus Collisions , 2004, hep-ph/0407264.

[3]  Space-time evolution of bulk QCD matter , 2006, nucl-th/0607018.

[4]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[5]  B. Betz,et al.  Complete second-order dissipative fluid dynamics , 2009 .

[6]  Y. Nara,et al.  Elliptic flow in Pb+Pb collisions at $\sqrt{s_{NN}} = 2.76$~TeV: hybrid model assessment of the first data , 2010, 1012.3955.

[7]  W. Israel Nonstationary irreversible thermodynamics: A Causal relativistic theory , 1976 .

[8]  M. Luzum,et al.  Conformal relativistic viscous hydrodynamics: Applications to RHIC results at s NN =200 GeV , 2008, 0804.4015.

[9]  S. Jeon,et al.  Elliptic and triangular flow in event-by-event D=3+1 viscous hydrodynamics. , 2010, Physical review letters.

[10]  J. Jia,et al.  Probing the properties of the strongly-interacting quark gluon plasma at RHIC , 2010 .

[11]  S. Bass,et al.  Triangular flow in event-by-event ideal hydrodynamics in Au+Au collisions at √SNN = 200A GeV , 2010, 1008.0625.

[12]  Ewald Müller,et al.  The analytical solution of the Riemann problem in relativistic hydrodynamics , 1994, Journal of Fluid Mechanics.

[13]  I. Muller,et al.  Zum Paradoxon der Warmeleitungstheorie , 1967 .

[14]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[15]  P. Bożek Flow and interferometry in (3 + 1)-dimensional viscous hydrodynamics , 2011, 1110.6742.

[16]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism , 1997 .

[17]  K. Eskola,et al.  Event-by-event hydrodynamics and elliptic flow from fluctuating initial states , 2010, 1007.0368.

[18]  Z. Fodor,et al.  Fluctuations of conserved charges at finite temperature from lattice QCD , 2011, 1112.4416.

[19]  C. Stivers Class , 2010 .

[20]  M. Luzum,et al.  Erratum: Conformal relativistic viscous hydrodynamics: Applications to RHIC results at {radical}(s{sub NN})=200 GeV [Phys. Rev. C 78, 034915 (2008)] , 2009 .

[21]  K. Dusling,et al.  Simulating elliptic flow with viscous hydrodynamics , 2007, 0710.5932.

[22]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[23]  A. H. Taub,et al.  Relativistic Rankine-Hugoniot Equations , 1948 .

[24]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[25]  Carlo Rovelli Quantum gravity , 2008, Scholarpedia.

[26]  Shu-ichiro Inutsuka,et al.  A fast numerical scheme for causal relativistic hydrodynamics with dissipation , 2011, J. Comput. Phys..

[27]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[28]  Quark gluon plasma and color glass condensate at RHIC? The Perspective from the BRAHMS experiment , 2005 .

[29]  Hadronic dissipative effects on elliptic flow in ultrarelativistic heavy-ion collisions , 2005, nucl-th/0511046.

[30]  I. Cordero-Carri'on,et al.  On the convexity of relativistic hydrodynamics , 2013, 1302.3758.

[31]  Werner Israel,et al.  Transient relativistic thermodynamics and kinetic theory , 1979 .

[32]  Exploring the vacuum geometry of N=1 gauge theories , 2006, hep-th/0604208.

[33]  J. Stewart,et al.  Thermodynamics of nonstationary and transient effects in a relativistic gas , 1976 .

[34]  The exact solution of the Riemann problem with non-zero tangential velocities in relativistic hydrodynamics , 2000, Journal of Fluid Mechanics.

[35]  Hans Christian Öttinger,et al.  General projection operator formalism for the dynamics and thermodynamics of complex fluids , 1998 .

[36]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[37]  K. Ohnishi,et al.  Derivation of covariant dissipative fluid dynamics in the renormalization-group method , 2007 .

[38]  L. Pang,et al.  Effects of initial flow velocity fluctuation in event-by-event (3+1)D hydrodynamics , 2012, 1205.5019.

[39]  P. Romatschke,et al.  Viscosity information from relativistic nuclear collisions: how perfect is the fluid observed at RHIC? , 2007, Physical review letters.

[40]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid , 1940 .

[41]  T. Ludlam Experimental results from the early measurements at RHIC; hunting for the quark–gluon plasma , 2005 .

[42]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[43]  A. Andronic,et al.  The thermal model on the verge of the ultimate test: particle production in Pb–Pb collisions at the LHC , 2011, 1106.6321.

[44]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. I. Development of a general formalism , 1997 .

[45]  V. Roy,et al.  2+1 dimensional hydrodynamics including bulk viscosity: A Systematics study , 2011, 1109.1630.

[46]  Zoltán Fodor,et al.  The QCD equation of state with dynamical quarks , 2010, 1007.2580.

[47]  J. Koenderink Q… , 2014, Les noms officiels des communes de Wallonie, de Bruxelles-Capitale et de la communaute germanophone.

[48]  M. Stephanov,et al.  Relativistic viscous hydrodynamics, conformal invariance, and holography , 2007, 0712.2451.

[49]  T. Kunihiro,et al.  Stable first-order particle-frame relativistic hydrodynamics for dissipative systems , 2007, 0709.3645.

[50]  Kenji Fukushima,et al.  The phase diagram of dense QCD , 2010, 1005.4814.

[51]  S. Bass,et al.  200 A GeV Au + Au collisions serve a nearly perfect quark-gluon liquid. , 2010, Physical review letters.

[52]  D. Rischke,et al.  Numerical tests of causal relativistic dissipative fluid dynamics , 2009, 0907.2583.