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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:



Code Basic characteristics


Roe type-l Riemann solver of Roe type with arithmetic averaging;
[179Jump To The Next Citation Point249Jump To The Next Citation Point93Jump To The Next Citation Point] monotonicity preserving, linear reconstruction of primitive variables;
second-order time stepping
([179Jump To The Next Citation Point249Jump To The Next Citation Point]: predictor-corrector; [93Jump To The Next Citation Point]: standard scheme).
Roe-Eulderink Linearized Riemann solver based on Roe averaging;
[83Jump To The Next Citation Point] second-order accuracy in space and time.
LCA-phm [176Jump To The Next Citation Point] Local linearization and decoupling of the system;
PHM reconstruction of characteristic fluxes;
third-order TVD preserving RK method for time stepping.
LCA-eno [74Jump To The Next Citation Point] 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 [181Jump To The Next Citation Point] 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
[89Jump To The Next Citation Point] equations in primitive form;
monotonic linear reconstruction of p, r, and ui;
second-order predictor-corrector time stepping.
MFF-ppm Marquina flux formula for numerical flux computation;
[183Jump To The Next Citation Point6Jump To The Next Citation Point] 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;
[75Jump To The Next Citation Point] upwind biased ENO/PHM reconstruction of characteristic fluxes;
second- and third-order TVD preserving RK methods for time stepping.
MFF-l [93Jump To The Next Citation Point] Marquina flux formula for numerical flux computation;
monotonic linear reconstruction of primitive variables;
standard second-order finite difference algorithms for time stepping.
Flux split [93Jump To The Next Citation Point] RTVD flux-split second-order method.
rGlimm [295Jump To The Next Citation Point] RGlimm’s method applied to RHD equations in primitive form;
first-order accuracy in space and time.
rBS [303Jump To The Next Citation Point] 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 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 [295Jump To The Next Citation Point].


Code Basic characteristics


RHLLE [256Jump To The Next Citation Point] Harten-Lax-van Leer approximate Riemann solver;
monotonic linear reconstruction of conserved/primitive variables;
second-order accuracy in space and time.
sTVD [138Jump To The Next Citation Point] Davis (1984) symmetric TVD scheme with nonlinear numerical dissipation;
second-order accuracy in space and time.
rAW [264] Global and local (first-order) and differential (second-order) artificial wind
methods.
sCENO [71Jump To The Next Citation Point] 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 [10Jump To The Next Citation Point] Non-oscillatory central difference scheme;
second-order accuracy in space (MUSCL-type piece-wise linear reconstruction)
and time (two step predictor corrector methods).


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


Code Basic characteristics


Artificial viscosity
AV-mono [50Jump To The Next Citation Point123Jump To The Next Citation Point187Jump To The Next Citation Point] 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 [213Jump To The Next Citation Point] Non-conservative formulation of the RHD equations;
internal energy equation;
consistent formulation of artificial viscosity;
adaptive mesh and implicit time stepping.
cAV-mono [10Jump To The Next Citation Point] 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 [77Jump To The Next Citation Point] Non-conservative formulation of the RHD equations
(transport differencing, equation for rhW);
explicit second-order Lax-Wendroff scheme with FCT algorithm.
SHASTA-c FCT algorithm based on SHASTA [33Jump To The Next Citation Point];
[256Jump To The Next Citation Point6970Jump To The Next Citation Point244Jump To The Next Citation Point246Jump To The Next Citation Point] advection of conserved variables.


van Putten’s approach
van Putten [287Jump To The Next Citation Point] 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;
[172Jump To The Next Citation Point] (SPH0), [150Jump To The Next Citation Point] artificial viscosity extra terms in momentum and energy equations;
second-order time stepping
([172Jump To The Next Citation Point]: predictor-corrector; [150Jump To The Next Citation Point]: RK method).
SPH-AV-1 [172Jump To The Next Citation Point] (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 [172Jump To The Next Citation Point] (SPH2) Total energy equation;
otherwise same as SPH-AV-1.
SPH-cAV-c [261Jump To The Next Citation Point] RHD equations in conservation form;
consistent formulation of artificial viscosity.
SPH-RS-c [53Jump To The Next Citation Point] RHD equations in conservation form;
dissipation terms constructed in analogy to terms in Riemann-
solver-based methods.
SPH-RS-gr [204Jump To The Next Citation Point] GR-SPH conservation equations [202Jump To The Next Citation Point];
dissipation terms as in [53Jump To The Next Citation Point].


Table 5: Code characteristics.


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