5 Summary of Methods 

This section contains a summary of all the methods reviewed in the two preceding sections 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 two tables (Table  2 for HRSC codes; Table  3 for other approaches).

 

Code	Basic characteristics
Roe type-l [107, 157, 59]	Riemann solver of Roe type with arithmetic averaging; monotonicity preserving, linear reconstruction of primitive variables; 2nd order time stepping ([107, 157]: predictor-corrector; [59]: standard scheme)
Roe-Eulderink [49]	Linearized Riemann solver based on Roe averaging; 2nd order accuracy in space and time
HLL-l [161]	Harten-Lax-van Leer approximate Riemann solver; monotonic linear reconstruction of conserved / primitive variables; 2nd order accuracy in space and time
LCA-phm [106]	Local linearization and decoupling of the system; PHM reconstruction of characteristic fluxes; 3rd order TVD preserving RK method for time stepping
LCA-eno [42]	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 [109]	Exact (ideal gas) Riemann solver; PPM reconstruction of primitive variables; 2nd order accuracy in time by averaging states in the domain of dependence of zone interfaces
Falle-Komissarov [55]	Approximate Riemann solver based on local linearizations of the RHD equations in primitive form; monotonic linear reconstruction of p, , and ; 2nd order predictor-corrector time stepping
MFF-ppm [111, 3]	Marquina flux formula for numerical flux computation; PPM reconstruction of primitive variables; 2nd and 3rd order TVD preserving RK methods for time stepping
MFF-eno/phm [43]	Marquina flux formula for numerical flux computation; upwind biased ENO/PHM reconstruction of characteristic fluxes; 2nd and 3rd order TVD preserving RK methods for time stepping
MFF-l [59]	Marquina flux formula for numerical flux computation; monotonic linear reconstruction of primitive variables; standard 2nd order finite difference algorithms for time stepping
Flux split [59]	TVD flux-split 2nd order method
sTVD [82]	Davis (1984) symmetric TVD scheme with nonlinear numerical dissipation; 2nd order accuracy in space and time
rGlimm [187]	Glimm's method applied to RHD equations in primitive form; 1st order accuracy in space and time
rBS [194]	Relativistic beam scheme solving equilibrium limit of relativistic Boltzmann equation; distribution function approximated by discrete beams of particles reproducing appropriate moments; 1st and 2nd order TVD, 2nd and 3rd order ENO schemes

  
Table 2: High-resolution shock-capturing methods. All the codes rely on a conservation form of the RHD equations with the exception of ref. [187].

 

Code	Basic characteristics
Artificial viscosity
AV-mono [28, 75, 113]	Non-conservative formulation of the RHD equations (transport differencing, internal energy equation); artificial viscosity extra term in the momentum flux; monotonic 2nd order transport differencing; explicit time stepping
cAV-implicit    [131]	Non-conservative formulation of the RHD equations; internal energy equation; consistent formulation of artificial viscosity; adaptive mesh and implicit time stepping
Flux corrected transport
FCT-lw [45]	Non-conservative formulation of the RHD equations (transport differencing, equation for ); explicit 2nd order Lax-Wendroff scheme with FCT algorithm
SHASTA-c [161, 39, 40]	FCT algorithm based on SHASTA [20]; advection of conserved variables
van Putten's approach
van Putten [181]	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 [102, 89] (SPH0)	Specific internal energy equation; artificial viscosity extra terms in momentum and energy equations; 2nd order time stepping ([102]: predictor-corrector; [89]: RK method)
SPH-AV-1 [102] (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 [102] (SPH2)	Total energy equation; otherwise same as SPH-AV-1
SPH-cAV-c [164]	RHD equations in conservation form; consistent formulation of artificial viscosity
SPH-RS-c [30]	RHD equations in conservation form; dissipation terms constructed in analogy to terms in Riemann solver based methods

  
Table 3: Code characteristics.

Numerical Hydrodynamics in Special Relativity Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de