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 [239Jump To The Next Citation Point]. Motivated by the search for a robust and accurate approximate Riemann solver that avoids these common failures, Donat and Marquina [76Jump To The Next Citation Point] have extended a numerical flux formula, which was first proposed by Shu and Osher [261Jump To The Next Citation Point] 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 [261Jump To The Next Citation Point]. 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 [76Jump To The Next Citation Point]. 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 [182Jump To The Next Citation Point183Jump To The Next Citation Point]. 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 [183Jump To The Next Citation Point]). 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 [75Jump To The Next Citation Point]. 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 [282Jump To The Next Citation Point]) and first-order accurate in time. Aloy et al. [6Jump To The Next Citation Point] have implemented the modified Marquina flux formula in their three-dimensional relativistic hydrodynamic code GENESIS. Font et al. [93Jump To The Next Citation Point] 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.

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