This section contains a summary of all the methods reviewed in the two preceding Sections 3 and 4 as well as several FCT and artificial viscosity codes. The main characteristic of the codes (dissipation algorithm, spatial and temporal orders of accuracy, reconstruction techniques) are listed in three table:

- Table 3 for HRSC codes using characteristic information,
- Table 4 for HRSC codes avoiding the use of such information, and
- Table 5 for other approaches.

Code | Basic characteristics |

Roe type-l | Riemann solver of Roe type with arithmetic averaging; |

[179, 250, 93] | monotonicity preserving, linear reconstruction of primitive variables; |

second-order time stepping | |

([179, 250]: predictor-corrector; [93]: standard scheme). | |

Roe–Eulderink | Linearized Riemann solver based on Roe averaging; |

[83] | second-order accuracy in space and time. |

LCA-phm [176] | Local linearization and decoupling of the system; |

PHM reconstruction of characteristic fluxes; | |

third-order TVD preserving RK method for time stepping. | |

LCA-eno [74] | Local linearization and decoupling of the system; |

high-order ENO reconstruction of characteristic split fluxes; | |

high-order TVD preserving RK methods for time stepping. | |

rPPM [181] | Exact (ideal gas) Riemann solver; |

PPM reconstruction of primitive variables; | |

second-order accuracy in time by averaging states in the domain of | |

dependence of zone interfaces. | |

Falle–Komissarov | Approximate Riemann solver based on local linearizations of the RHD |

[89] | equations in primitive form; |

monotonic linear reconstruction of , , and ; | |

second-order predictor-corrector time stepping. | |

MFF-ppm | Marquina flux formula for numerical flux computation; |

[183, 6] | PPM reconstruction of primitive variables; |

second- and third-order TVD preserving RK methods for time stepping. | |

MFF-eno/phm | Marquina flux formula for numerical flux computation; |

[75] | upwind biased ENO/PHM reconstruction of characteristic fluxes; |

second- and third-order TVD preserving RK methods for time stepping. | |

MFF-l [93] | Marquina flux formula for numerical flux computation; |

monotonic linear reconstruction of primitive variables; | |

standard second-order finite difference algorithms for time stepping. | |

Flux split [93] | RTVD flux-split second-order method. |

rGlimm [295] | RGlimm’s method applied to RHD equations in primitive form; |

first-order accuracy in space and time. | |

rBS [303] | Relativistic beam scheme solving equilibrium limit of relativistic |

Boltzmann equation; | |

distribution function approximated by discrete beams of particles | |

reproducing appropriate moments; | |

first- and second-order TVD, second-order and third-order ENO schemes. | |

Code | Basic characteristics |

RHLLE [257] | Harten–Lax–van Leer approximate Riemann solver; |

monotonic linear reconstruction of conserved/primitive variables; | |

second-order accuracy in space and time. | |

sTVD [138] | Davis (1984) symmetric TVD scheme with nonlinear numerical dissipation; |

second-order accuracy in space and time. | |

rAW [265] | Global and local (first-order) and differential (second-order) artificial wind |

methods. | |

sCENO [71] | Symmetric first-order numerical flux (HLL, local Lax–Friedrichs); |

high-order (convex) ENO interpolation; | |

second-order and third-order TVD preserving RK methods for time stepping. | |

NOCD [10] | Non-oscillatory central difference scheme; |

second-order accuracy in space (MUSCL-type piece-wise linear reconstruction) | |

and time (two step predictor corrector methods). | |

Code | Basic characteristics |

Artificial viscosity | |

AV-mono [50, 123, 187] | Non-conservative formulation of the RHD equations |

(transport differencing, internal energy equation); | |

artificial viscosity extra term in the momentum flux; | |

monotonic second-order transport differencing; | |

explicit time stepping. | |

cAV-implicit [214] | Non-conservative formulation of the RHD equations; |

internal energy equation; | |

consistent formulation of artificial viscosity; | |

adaptive mesh and implicit time stepping. | |

cAV-mono [10] | Non-conservative formulation of the RHD equations |

(transport differencing, internal energy equation); | |

consistent bulk scalar and tensorial artificial viscosity; | |

monotonic second-order transport differencing; | |

explicit time stepping. | |

Flux corrected transport | |

FCT-lw [77] | Non-conservative formulation of the RHD equations |

(transport differencing, equation for ); | |

explicit second-order Lax–Wendroff scheme with FCT algorithm. | |

SHASTA-c | FCT algorithm based on SHASTA [33]; |

[257, 69, 70, 245, 247] | advection of conserved variables. |

van Putten’s approach | |

van Putten [287] | Ideal RMHD equations in constraint-free, divergence form; |

evolution of integrated variational parts of conserved quantities; | |

smoothing algorithm in numerical differentiation step; | |

leap-frog method for time stepping. | |

Smooth particle hydrodynamics | |

SPH-AV-0 | Specific internal energy equation; |

[172] (SPH0), [150] | artificial viscosity extra terms in momentum and energy equations; |

second-order time stepping | |

([172]: predictor-corrector; [150]: RK method). | |

SPH-AV-1 [172] (SPH1) | Time derivatives in SPH equations include variations in smoothing |

length and mass per particle; | |

Lorentz factor terms treated more consistently; | |

otherwise same as SPH-AV-0. | |

SPH-AV-c [172] (SPH2) | Total energy equation; |

otherwise same as SPH-AV-1. | |

SPH-cAV-c [262] | RHD equations in conservation form; |

consistent formulation of artificial viscosity. | |

SPH-RS-c [53] | RHD equations in conservation form; |

dissipation terms constructed in analogy to terms in Riemann- | |

solver-based methods. | |

SPH-RS-gr [204] | GR-SPH conservation equations [202]; |

dissipation terms as in [53]. | |

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