v 1 3 1 M ay 2 00 6 REFINING A RELATIVISTIC , HYDRODYNAMIC SOLVER : ADMITTING ULTRA - RELATIVISTIC FLOWS ∗

We have undertaken the simulation of hydrodynamic flows with bulk Lorentz factors in the range 10 2 –10 6. We discuss the application of an existing relativistic, hydrodynamic primitive-variable recovery algorithm to a study of pulsar winds, and, in particular, the refinement necessary to admit such ultra-relativistic flows. We show that the use of an analytical quartic root finder is required for Lorentz factors above 10 2 , but that an iterative quartic root finder, which is known to be robust for Lorentz factors up to at least 25, offers a 24% speed advantage. We demonstrate the existence of a simple diagnostic allowing for a hybrid primitives recovery algorithm that includes an automatic, real-time toggle between the iterative and analytical methods. We further determine the accuracy of the iterative and hybrid algorithms for a comprehensive selection of input parameters. 1. Introduction. Hydrodynamic simulations have been widely used to model a broad range of physical systems. When the velocities involved are a small fraction of the speed of light and gravity is weak, the classical Newtonian approximation to the equations of motion may be used. However, these two conditions are violated for a host of interesting scenarios, including, for example, heavy ion collision systems [6], relativistic laser systems [3], and many from astrophysics [9] (and references therein), that call for a fully relativistic, hydrodynamic (RHD) treatment. The methods of solution of classical hydrodynamic problems have been successfully adapted to those of a RHD nature, albeit giving rise to significant complication; in particular, the physical quantities of a hydrodynamic flow (the rest-frame mass density, n, pressure, p, and velocity, v) are coupled to the conserved quantities (the laboratory-frame mass density, R, momentum density, M , and energy density, E) via the Lorentz transformation. The fact that modern RHD codes typically evolve the conserved quantities necessitates the recovery of the physical quantities (often referred to as the " primitive variables ") from the conserved quantities in order to obtain the flow velocity. Thus, the calculation of the primitives from the conserved variables has become a critical element of modern RHD codes [10]. In this paper, we present a method for recovering the primitive variables from the conserved quantities representing special relativistic, hydrodynamic (SRHD) flows with bulk Lorentz factors (γ = (1 − v 2) −1/2 , where v is the bulk flow velocity normalized to the speed of light) up to 10 6. …