The non-relativistic scheme of Sanders & Prendergast [159] has been extended to relativistic flows by Yang et al. [194]. They replaced the Maxwellian distribution function by its relativistic analogue, i.e., by the more complex Jüttner distribution function, which involves modified Bessel functions. For three-dimensional flows the Jüttner distribution function is approximated by seven delta functions or discrete beams of particles, which can be viewed as dividing the particles in each cell into seven distinct groups. In the local rest frame of the cell these seven groups represent particles at rest and particles moving in and directions, respectively.

Yang et al. [194] show that the integration scheme for the beams can be cast in the form of an upwind conservation scheme in terms of numerical fluxes. They further show that the beam scheme not only splits the state vector but also the flux vectors, and has some entropy-satisfying mechanism embedded as compared with approximate relativistic Riemann solver [42, 161] based on Roe's method [155]. The simplest relativistic beam scheme is only first-order accurate in space, but can be extended to higher-order accuracy in a straightforward manner. Yang et al. consider three high-order accurate variants (TVD2, ENO2, ENO3) generalizing their approach developed in [195, 196] for Newtonian gas dynamics, which is based on the essentially non-oscillatory (ENO) piecewise polynomial reconstruction scheme of Harten at al. [73].

Yang et al. [194] present several numerical experiments including relativistic one-dimensional shock tube flows and the simulation of relativistic two-dimensional Kelvin-Helmholtz instabilities. The shock tube experiments consist of a mildly relativistic shock tube, relativistic shock heating of a cold flow, the relativistic blast wave interaction of Woodward & Colella [191] (see Section 6.2.3), and the perturbed relativistic shock tube flow of Shu & Osher [163].

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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