A relativistic version of Davis’ method has been
used by Koide et al. [138, 136
, 210
] in 2D and 3D
simulations of relativistic magneto-hydrodynamic jets with moderate
Lorentz factors. Although the results obtained are encouraging, the
coarse grid zoning used in these simulations and the relative
smallness of the beam flow Lorentz factor (4.56, beam speed
) does not allow for a comparison with
Riemann-solver-based HRSC methods in the ultra-relativistic
limit.
Davis’ method is second-order accurate in space
and time. However, when simulating complex hydrodynamic and
especially magneto-hydrodynamic flows, accuracy is an important
issue. To this end Del Zanna and Bucciantini [71] have presented a
global third order accurate, centered scheme for multi-dimensional
SRHD. The basic properties of Del Zanna and Bucciantini’s method
are based on the work of Liu and Osher [164
]:
To preserve the symmetric property of the method, monotonic high-order numerical fluxes are computed at zone interfaces by means of central-type Riemann solvers avoiding spectral decomposition (e.g., Lax-Friedrichs numerical flux). The authors also test the Riemann solver of Harten, Lax, and van Leer within the framework of non-biased Riemann solvers.
Recently, Anninos and Fragile [10] have developed a
second order, non-oscillatory, central difference (NOCD) scheme for
the numerical integration of the GRHD equations. The code uses
MUSCL-type piecewise linear spatial interpolation to achieve
second-order accuracy in space. Second-order accuracy in time is
guaranteed by means of a predictor-corrector procedure. Symmetric
numerical fluxes are evaluated after the predictor step. The
results obtained in a series of challenging test problems (see
Section 6) are encouraging.