] proposed an explicit scheme to solve the equilibrium limit of
the non-relativistic Boltzmann equation, i.e., the Euler
equations of Newtonian fluid dynamics. In their so-called beam
scheme the Maxwellian velocity distribution function is
approximated by several Dirac delta functions or discrete beams
of particles in each computational cell, which reproduce the
appropriate moments of the distribution function. The beams
transport mass, momentum, and energy into adjacent cells, and
their motion is followed to first-order accuracy. The new (i.e.,
time advanced) macroscopic moments of the distribution function
are used to determine the new local non-relativistic Maxwell
distribution in each cell. The entire process is then repeated
for the next time step. The Courant-Friedrichs-Levy (CFL)
stability condition requires that no beam of gas travels farther
than one cell in one time step. This beam scheme, although being
a particle method derived from a microscopic kinetic description,
has all the desirable properties of modern characteristic-based
wave propagating methods based on a macroscopic continuum
description.
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].
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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |