Summary of Methods

5 Summary of Methods

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.

Table 3: High-resolution shock-capturing methods using characteristic information. All the codes rely on a conservation form of the RHD equations with the exception of [295

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 $p$ , $ρ$ , and $i u$ ;
	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.

Table 4: High-resolution shock-capturing methods avoiding the use of characteristic information.

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).

Table 5: Code characteristics.

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 $ρhW$ );
	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 ].