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 made to admit such ultra-relativistic flows. We show that an iterative quartic root finder breaks down for Lorentz factors above 10^2 and employ an analytic root finder as a solution. We find that the former, which is known to be robust for Lorentz factors up to at least 50, 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 and demonstrate the latters capability to elucidate the internal structure of ultra-relativistic plasmas. In particular, we discuss simulations showing that the interaction of a light, ultra-relativistic pulsar wind with a slow, dense ambient medium can give rise to asymmetry reminiscent of the Guitar nebula leading to the formation of a relativistic backflow harboring a series of internal shockwaves. The shockwaves provide thermalized energy that is available for the continued inflation of the PWN bubble. In turn, the bubble enhances the asymmetry, thereby providing positive feedback to the backflow.
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