6 Test BenchNumerical Hydrodynamics in Special Relativity4.3 Relativistic beam scheme

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
[107Jump To The Next Citation Point In The Article, 157Jump To The Next Citation Point In The Article, 59Jump To The Next Citation Point In The Article]
Riemann solver of Roe type with arithmetic averaging; monotonicity preserving, linear reconstruction of primitive variables; 2nd order time stepping ([107Jump To The Next Citation Point In The Article, 157Jump To The Next Citation Point In The Article]: predictor-corrector; [59Jump To The Next Citation Point In The Article]: standard scheme)
Roe-Eulderink
[49Jump To The Next Citation Point In The Article]
Linearized Riemann solver based on Roe averaging; 2nd order accuracy in space and time
HLL-l
[161Jump To The Next Citation Point In The Article]
Harten-Lax-van Leer approximate Riemann solver; monotonic linear reconstruction of conserved / primitive variables; 2nd order accuracy in space and time
LCA-phm
[106Jump To The Next Citation Point In The Article]
Local linearization and decoupling of the system; PHM reconstruction of characteristic fluxes; 3rd order TVD preserving RK method for time stepping
LCA-eno
[42Jump To The Next Citation Point In The Article]
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
[109Jump To The Next Citation Point In The Article]
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
[55Jump To The Next Citation Point In The Article]
Approximate Riemann solver based on local linearizations of the RHD equations in primitive form; monotonic linear reconstruction of p, tex2html_wrap_inline5637, and tex2html_wrap_inline6185 ; 2nd order predictor-corrector time stepping
MFF-ppm
[111Jump To The Next Citation Point In The Article, 3Jump To The Next Citation Point In The Article]
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
[43Jump To The Next Citation Point In The Article]
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
[59Jump To The Next Citation Point In The Article]
Marquina flux formula for numerical flux computation; monotonic linear reconstruction of primitive variables; standard 2nd order finite difference algorithms for time stepping
Flux split
[59Jump To The Next Citation Point In The Article]
TVD flux-split 2nd order method
sTVD
[82Jump To The Next Citation Point In The Article]
Davis (1984) symmetric TVD scheme with nonlinear numerical dissipation; 2nd order accuracy in space and time
rGlimm
[187Jump To The Next Citation Point In The Article]
Glimm's method applied to RHD equations in primitive form; 1st order accuracy in space and time
rBS
[194Jump To The Next Citation Point In The Article]
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. [187Jump To The Next Citation Point In The Article].

 

Code Basic characteristics
Artificial viscosity
AV-mono
[28Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article, 113Jump To The Next Citation Point In The Article]
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   
[131Jump To The Next Citation Point In The Article]
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
[45Jump To The Next Citation Point In The Article]
Non-conservative formulation of the RHD equations (transport differencing, equation for tex2html_wrap_inline6187); explicit 2nd order Lax-Wendroff scheme with FCT algorithm
SHASTA-c
[161Jump To The Next Citation Point In The Article, 39, 40Jump To The Next Citation Point In The Article]
FCT algorithm based on SHASTA
[20Jump To The Next Citation Point In The Article]; advection of conserved variables
van Putten's approach
van Putten
[181Jump To The Next Citation Point In The Article]
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
[102Jump To The Next Citation Point In The Article, 89Jump To The Next Citation Point In The Article] (SPH0)
Specific internal energy equation; artificial viscosity extra terms in momentum and energy equations; 2nd order time stepping ([102Jump To The Next Citation Point In The Article]: predictor-corrector; [89Jump To The Next Citation Point In The Article]: RK method)
SPH-AV-1
[102Jump To The Next Citation Point In The Article] (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
[102Jump To The Next Citation Point In The Article] (SPH2)
Total energy equation; otherwise same as SPH-AV-1
SPH-cAV-c
[164Jump To The Next Citation Point In The Article]
RHD equations in conservation form; consistent formulation of artificial viscosity
SPH-RS-c
[30Jump To The Next Citation Point In The Article]
RHD equations in conservation form; dissipation terms constructed in analogy to terms in Riemann solver based methods
  
Table 3: Code characteristics.








6 Test BenchNumerical Hydrodynamics in Special Relativity4.3 Relativistic beam scheme

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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