A relativistic version of Davis’ method has been used by Koide et al. [138, 136, 211] 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 0.98 c) 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  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 :
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  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.
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