Godunov-type schemes are indeed very robust in most situations
although they fail spectacularly on occasions. Reports on
approximate Riemann solver failures and their respective
corrections (usually a judicious addition of artificial
dissipation) are abundant in the literature . Motivated by the search for a robust and accurate approximate
Riemann solver that avoids these common failures, Donat &
Marquina  have extended to systems a numerical flux formula which was
first proposed by Shu & Osher  for scalar equations. In the scalar case and for characteristic
wave speeds which do not change sign at the given numerical
interface, Marquina's flux formula is identical to Roe's flux.
Otherwise, the scheme switches to the more viscous, entropy
satisfying local Lax-Friedrichs scheme . In the case of systems, the combination of Roe and
local-Lax-Friedrichs solvers is carried out in each
characteristic field after the local linearization and decoupling
of the system of equations . However, contrary to Roe's and other linearized methods, the
extension of Marquina's method to systems is not based on any
averaged intermediate state.
Martí et al. have used this method in their simulations of
relativistic jets [110,
111]. The resulting numerical code has been successfully used to
describe ultra-relativistic flows in both one and two spatial
dimensions with great accuracy (a large set of test calculations
using Marquina's Riemann solver can be found in Appendix II
of ). Numerical experimentation in two dimensions confirms that the
dissipation of the scheme is sufficient to eliminate the
carbuncle phenomenon , which appears in high Mach number relativistic jet simulations
when using other standard solvers .
Aloy et al.  have implemented Marquina's flux formula in their three
dimensional relativistic hydrodynamic code GENESIS.
Font et al.  have developed a 3D general relativistic hydro code where the
matter equations are integrated in conservation form and fluxes
are calculated with Marquina's formula.
||Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351