Overview of the numerical methods

1.2 Overview of the numerical methods

The first attempt to solve the equations of relativistic hydrodynamics (RHD) was made by Wilson [299 , 296 ] and collaborators [50

, 123

] using an Eulerian explicit finite difference code with monotonic transport. The code relies on artificial viscosity techniques [293 , 244

] to handle shock waves. It has been widely used to simulate flows encountered in cosmology, axisymmetric relativistic stellar collapse, accretion onto compact objects and, more recently, collisions of heavy ions. Almost all the codes for both special (SRHD) and general (GRHD) numerical relativistic hydrodynamics developed in the eighties [224 , 268 , 208 , 206 , 209 , 85 ] were based on Wilson’s procedure. However, despite its popularity it turned out to be unable to accurately describe extremely relativistic flows (Lorentz factors larger than 2; see, e.g., [50

]).

In the mid-eighties, Norman and Winkler [214 ] proposed a reformulation of the difference equations of SRHD with an artificial viscosity consistent with the relativistic dynamics of non-perfect fluids. The strong coupling introduced in the equations by the presence of the viscous terms in the definition of relativistic momentum and total energy densities required an implicit treatment of the difference equations. Accurate results across strong relativistic shocks with large Lorentz factors were obtained in combination with adaptive mesh techniques. However, no multi-dimensional version of this code was developed.

Attempts to integrate the RHD equations avoiding the use of artificial viscosity were performed in the early nineties. Dubal [77 ] developed a 2D code for relativistic magneto-hydrodynamics based on an explicit second-order Lax–Wendroff scheme incorporating a flux-corrected transport (FCT) algorithm [33 ]. Following a completely different approach Mann [172 ] proposed a multi-dimensional code for GRHD based on smoothed particle hydrodynamics (SPH) techniques [199 ], which he applied to relativistic spherical collapse [174 ]. When tested against 1D relativistic shock tubes all these codes performed similar to the code of Wilson. More recently, Dean et al. [69 ] have applied flux correcting algorithms for the SRHD equations in the context of heavy ion collisions. Recent developments in relativistic SPH methods [53 , 262 ] are discussed in Section 4.2.

A major breakthrough in the simulation of ultra-relativistic flows was accomplished when high-resolution shock-capturing (HRSC) methods, specially designed to solve hyperbolic systems of conservations laws, were applied to solve the SRHD equations [179 , 176 , 83 , 84 ].