### 3.8 Marquina’s flux formula

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 [239]. Motivated by the
search for a robust and accurate approximate Riemann solver that avoids these common failures,
Donat and Marquina [76] have extended a numerical flux formula, which was first proposed by
Shu and Osher [261] for scalar equations, to systems of 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 [261]. 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 [76]. 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 a version of Marquina’s method that applies the Lax–Friedrichs flux to all
fields (modified Marquina’s flux formula) in their simulations of relativistic jets [182, 183].
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 [183]). Numerical experimentation
in two dimensions confirms that the dissipation of the scheme is sufficient to eliminate the
carbuncle phenomenon [239], which appears in high Mach number relativistic jet simulations when
using other standard solvers [75]. 2D Simulations of relativistic AGN jets using Marquina’s flux
formula have also been performed by Mizuta et al. [196], the code being second-order accurate
in space (MUSCL reconstruction [282]) and first-order accurate in time. Aloy et al. [6] have
implemented the modified Marquina flux formula in their three-dimensional relativistic hydrodynamic
code GENESIS. Font et al. [93] 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.