The non-relativistic scheme of Sanders and
Prendergast [253] has been extended to
relativistic flows by Yang et al. [303]. 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 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. [303] show that the
integration scheme for the beams can be cast into 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 an approximate relativistic
Riemann solver [74
, 256
] based on Roe’s
method [247]. 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 [304, 305] for Newtonian gas
dynamics, which is based on the essentially non-oscillatory (ENO)
piecewise polynomial reconstruction scheme of Harten et
al. [120
].
Yang et al. [303] 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
and Colella [300
] (see Section
6.2.3), and the perturbed relativistic
shock tube flow of Shu and Osher [260
].