A survey is given on the applications of hydrodynamic model of nucleus-nucleus collisons, focusing especially on i) the resolution of hydrodynamic equations for arbitrary configurations, by using the smoothed-particle hydrodynamic approach; ii) effects of the event-by-event fluctuation of the initial conditions on the observables; iii) decoupling criteria; iv) analytical solutions; and others.

The effects of fluctuating initial conditions are studied in the context of relativistic heavy ion collisions where a rapidly evolving system is formed. Two-particle correlation analysis is applied to events generated with the NEXSPHERIO hydrodynamic code, starting with fluctuating nonsmooth initial conditions (IC). The results show that the nonsmoothness in the IC survives the hydroevolution and can be seen as topological features of the angular correlation function of the particles emerging from the evolving system. A long range correlation is observed in the longitudinal direction and in the azimuthal direction a double peak structure is observed in the opposite direction to the trigger particle. This analysis provides clear evidence that these are signatures of the combined effect of tubular structures present in the IC and the proceeding collective dynamics of the hot and dense medium.

A new method for evaluating spectra and correlations in the hydrodynamic approach is proposed. It is based on an analysis of the Boltzmann equations (BE) in terms of probabilities for constituent particles to escape from the interacting system. The conditions of applicability of the Cooper-Frye freeze-out prescription are considered within the method. The results are illustrated with a nonrelativistic exact solution of BE for an expanding spherical fireball as well as with approximate solutions for ellipsoidally expanding ones.

Elliptic flow at BNL RHIC is computed event by event with NeXSPheRIO. We show that when symmetry of the particle distribution in relation to the reaction plane is assumed, as usually done in the experimental extraction of elliptic flow, there is a disagreement between the true and reconstructed elliptic flows (15%-30% for eta=0, 30% for p perpendicular=0.5 GeV). We suggest a possible way to take into account the asymmetry and get good agreement between these elliptic flows.

The method of smoothed particle hydrodynamics (SPH) is developped appropriately for the study of relativistic heavy ion collision processes. In order to describe the flow of a high energy but low baryon number density fluid, the entropy is taken as the SPH base. We formulate the method in terms of the variational principle. Several examples show that the method is very promising for the study of hadronic flow in RHIC physics. I. INTRODUCTIONHydrodynamic descriptions of high energy hadronic and nuclear collisions have a rather long history [1]. Although, from theoretical point of view, it is not a trivial matter to justify their validity, they have been successful in reproducing certain features of these processes, such as the energy dependence of the average multiplicity and the transverse-energy distributions. More recently, relativistic fluid dynamics have become an important tool for the analysis of relativistic heavy-ion-collision processes (for example, [2][3][4] and references there in). In these processes, the nuclear matter is expected to be compressed and heated up close to those states of the matter realized in the Big Bang Era of the Universe. Some laboratory data of these fascinating processes have already been obtained in the series of CERN experiments [5], and data at even higher temperatures and densities will soon be available in the forthcoming RHIC experiments.The relativistic hydrodynamics is a description based on local conservation laws, together with the hypothesis of local thermodynamical equilibrium. The conservation laws are written in terms of the four-divergence of the energymomentum tensor. The resulting system of equations is highly nonlinear and analytical solutions are only available for some very particular and limited configurations and equations of state [1,6]. Thus numerical approaches are resorted to but they usually depend on some sophisticated techniques specific to some symmetry involved in the problem. When no symmetry is involved, these methods become computationally very expensive. However this is exactly the case when we are challenged to a realistic three dimensional simulation for nuclear-collision processes. There, we expect no geometrical symmetry so that a full 3D calculation is required.One basic point in the hydrodynamic approach of relativistic nuclear collisions is that its principal ingredients, i.e., the equation of state of the matter and the initial conditions for the dynamics are not quite well known. On the contrary, we apply the hydrodynamic models to infer these informations on the properties of the matter in such a highly condensed and excited state. Thus we need to perform many hydrodynamic-model calculations for different equations of state and initial conditions to compare with the experimental data. In such a process, we actually don't need the very precise solution of hydrodynamic equations, but a general flow pattern which characterizes the final configuration of the system as a response to a given set of equation of state and initial conditions. ...