### 3.9 Symmetric TVD, ENO schemes with nonlinear numerical dissipation

The methods discussed in Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, and 3.8 are all based on exact or
approximate solutions of Riemann problems at cell interfaces in order to stabilize the discretization scheme
across strong shocks. Another successful approach relies on the addition of nonlinear dissipation terms to
standard finite difference methods. The algorithm of Davis [68] is based on such an approach. It can be
interpreted as a Lax–Wendroff scheme with a conservative TVD dissipation term. The numerical dissipation
term is local, free of problem dependent parameters and does not require any characteristic information.
This last fact makes the algorithm extremely simple when applied to any hyperbolic system of conservation
laws.
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 [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]:

- the use of point values instead of cell averages,
- time integration with TVD Runge–Kutta methods, and
- third-order accurate ENO reconstruction algorithm.

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.