3.8 Symmetric TVD schemes with 3 High-Resolution Shock-Capturing Methods3.6 Relativistic HLL method

3.7 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 [153Jump To The Next Citation Point In The Article]. Motivated by the search for a robust and accurate approximate Riemann solver that avoids these common failures, Donat & Marquina [44Jump To The Next Citation Point In The Article] have extended to systems a numerical flux formula which was first proposed by Shu & Osher [163Jump To The Next Citation Point In The Article] 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 [163Jump To The Next Citation Point In The Article]. 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 [44Jump To The Next Citation Point In The Article]. 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 [110Jump To The Next Citation Point In The Article, 111Jump To The Next Citation Point In The Article]. 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 [111Jump To The Next Citation Point In The Article]). Numerical experimentation in two dimensions confirms that the dissipation of the scheme is sufficient to eliminate the carbuncle phenomenon [153], which appears in high Mach number relativistic jet simulations when using other standard solvers [43Jump To The Next Citation Point In The Article].

Aloy et al. [3Jump To The Next Citation Point In The Article] have implemented Marquina's flux formula in their three dimensional relativistic hydrodynamic code GENESIS.

Font et al. [59Jump To The Next Citation Point In The Article] 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.



3.8 Symmetric TVD schemes with 3 High-Resolution Shock-Capturing Methods3.6 Relativistic HLL method

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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