Marquina’s flux formula

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.