High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations.The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory(WENO)technique as well as explicit Runge-Kutta time discretization.The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair.As soon as the entropy conservative flux is derived,the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function.The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

AnAdaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics

This paper extends the adaptive moving mesh method developed by Tang and Tang[36]to two-dimensional(2D)relativistic hydrodynamic(RHD)equations.The algorithm consists of two“independent”parts:the time evolution of the RHD equations and the(static)mesh iteration redistribution.In the first part,the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations.The second part is an iterative procedure.In each iteration,the mesh points are first redistributed,and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way.Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.

TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS

This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38(2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed.With the aid of the direct Eulerian GRP(generalized Riemann problem)methods and the analytical resolution of the local“quasi 1D”GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations.Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

Genuinely Multidimensional Physical-Constraints-Preserving FiniteVolume Schemes for the Special Relativistic Hydrodynamics

This paper develops the genuinely multidimensional HLL Riemann solver for the two-dimensional special relativistic hydrodynamic equations on Cartesian meshes and studies its physical-constraint-preserving(PCP)property.Based on the resulting HLL solver,the first-and high-order accurate PCP finite volume schemes are proposed.In the high-order scheme,the WENO reconstruction,the third-order accurate strong-stability-preserving time discretizations and the PCP flux limiter are used.Several numerical results are given to demonstrate the accuracy,performance and resolution of the shock waves and the genuinely multi-dimensional wave structures etc.of our PCP finite volume schemes.

Effects of bulk viscosity on hadron spectra and the Hanbury-Brown Twiss radius by causal viscous hydrodynamics

The particle spectra and Hanbury-Brown Twiss （HBT） radius of Au＋Au collisions at RHIC energy are investigated by a hydrodynamical expanding source with both shear and bulk viscosities （ζ）. With a large width of the ratio of ζ to entropy density s, both the particle transverse momentum spectra and the ratio Rout/Raide Of HBT radii in the direction of the total transverse momentum of detected two particles （Rout） and perpendicular to both this direction and the beam direction （Rside） become a little steeper.

Numerical Method for Solving Relativistic Hydrodynamic Equation Including Quark-Fragment Effect

In this paper, a numerical method is given to solve relativistic hydrodynamic equations with source terms by conservative finite difference scheme. In calculation, QGP (quark gluon plasma) phase transition is also considered. The numerical experiments have verified the effectiveness of the numerical method and some computational results are illustrated.

Signals of α clusters in ^(16)O+^(16)O collisions at the LHC from relativistic hydrodynamic simulations

In relativistic heavy ion collisions,the fluctuations of initial entropy density convert to the correlations of final state hadrons in momentum space through the collective expansion of strongly interacting QCD matter.Using a(3+1)D viscous hydrodynamic program,CL Visc,we consider whether the nuclear structure,which provides initial state fluctuations as well as correlations,can affect the final state of heavy ion collisions,and whether one can find signals of α cluster structures in oxygen using final state observables in ^(16)O+ ^(16)O collisions at the CERN Large Hadron Collider.For the initial nucleon distributions in oxygen nuclei,we compare three different configurations,a tetrahedral structure with four-α clusters,the deformed Woods-Saxon distribution,and a spherical symmetric Woods-Saxon distribution.Our results show that the charged multiplicity as a function of centrality and the elliptic flow at the most central collisions using the four-α structure differs from those with the Woods-Saxon and deformed Woods-Saxon distributions,which may help to identify α clustering structures in oxygen nuclei.

Longitudinal dynamics from hydrodynamics with an order parameter

We studied coupled dynamics of hydrodynamic fields and order parameter in the presence of nontrivial longitudinal flow using the chiral fluid dynamics model.We found that longitudinal expansion provides an effective relaxation for the order parameter,which equilibrates in an oscillatory fashion.Similar oscillations are also visible in hydrodynamic degrees of freedom through coupled dynamics.The oscillations are reduced when dissipation is present.We also found that the quark density,which initially peaked at the boundary of the boost invariant region,evolves toward forward rapidity with the peak velocity correlated with the velocity of longitudinal expansion.The peak broadens during this evolution.The corresponding chemical potential rises due to simultaneous decrease of density and temperature.We compared the cases with and without dissipation for the order parameter and also the standard hydrodynamics without order parameter.We found that the corresponding effects on temperature and chemical potential can be understood from the conservation laws and different speeds of equilibration of the order parameter in the three cases.

Bjorken Flow Revisited:Analytic and Numerical Solutions in Flat Space-Time Coordinates

In this work we provide analytic and numerical solutions for the Bjorken flow,a standard benchmark in relativistic hydrodynamics providing a simple model for the bulk evolution of matter created in collisions between heavy nuclei.We consider relativistic gases of both massive and massless particles,working in a(2+1)and(3+1)Minkowski space-time coordinate system.The numerical results from a recently developed lattice kinetic scheme show excellent agreement with the analytic solutions.

Hydrodynamic description of D meson production in high-energy heavy-ion collisions

The large values and constituent-quark-number scaling of the elliptic flow of low-pT D mesons imply that charm quarks,initially produced through hard processes,might be partially thermalized through strong interactions with quark-gluon plasma(QGP)in high-energy heavy-ion collisions.To quantify the degree of thermalization of low-pT charm quarks,we compare the D^(0)meson spectra and elliptic flow from a hydrodynamic model to experimental data as well as transport model simulations.We use an effective charm chemical potential at the freeze-out temperature to account for the initial charm quark production from hard processes and assume that they are thermalized in the local comoving frame of the medium before freeze-out.D^(0)mesons are sampled statistically from the freeze-out hyper-surface of the expanding QGP as described by the event-by-event(3+1)D viscous hydrodynamic model CLVisc.Both the hydrodynamic and transport models can describe the elliptic flow of D^(0)mesons at pT~3 GeV/c as measured in Au+Au collisions at√SNN=200 GeV.Though the experimental data on D^(0)spectra are consistent with the hydrodynamic result at small pT~1 GeV/c,they deviate from the hydrodynamic model at high transverse momentum,pT~2 GeV/c.The diffusion and parton energy loss mechanisms in the transport model can describe the measured spectra reasonably well within the theoretical uncertainty.Our comparative study indicates that charm quarks only approach local thermal equilibrium at small pT,even though they acquire sizable elliptic flow comparable to light-quark hadrons at both small and intermediate pT.

On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.