3.8 Symmetric TVD schemes with nonlinear numerical dissipation 

The methods discussed in the previous subsections 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 [38] is based on such an approach. It can be interpreted as a Lax-Wendroff scheme with a conservative TVD (total variation diminishing) 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. [82, 81, 129] 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.

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