Table 2.

State-of-the-art GRMHD multidimensional numerical codes for test fluid evolutions.

Code	Main features
	Full-wave decomposition Riemann Solver HRSC
K05 [199, 201]	2D. Special relativistic Riemann solver extended to GRMHD following the approach of [320]. Right eigenvectors in primitive variables. No left eigenvectors. Jumps in characteristic variables from physical conditions across discontinuities. Switch between eigenvector sets for degenerate and nondegenerate states. Extra artificial viscosity and resistivity. 2nd-order accuracy in space and time. CT scheme for divergence-free constraint.
A06 [24]	2D. Right and left eigenvectors in conserved variables from covariant ones. Single set of left/right eigenvectors for both degenerate and nondegenerate states. 2nd-order accuracy in space and time. Extension to GRMHD follows the approach of [320]. Flux-CT scheme for divergence-free constraint.
	Symmetric HRSC / Incomplete Riemann solvers
K98 [196]	2D. Symmetric TVD scheme with nonlinear numerical dissipation [83]. 2nd-order accuracy in space and time. No method to preserve the magnetic field divergence.
HARM [149]	2D. Kerr-Schild-type coordinates (Kerr black-hole spacetime). HLL approximate Riemann solver. MC, minmod, and van Leer slope limiters. Flux-CT scheme for divergence-free constraint. Approximation for maximum wave speeds (quadratic equation). 2nd-order accuracy in space and time.
cosmos++ [20]	3D. Conservative energy formulation. Kurganov-Tadmor central scheme. Minmod, van Leer, and superbee slope limiters. Unstructured grids and local AMR capabilities. Divergence cleaning of magnetic field constraint (parabolic or hyperbolic). 2nd-order accuracy.
raishin [265]	3D. Follows conservative formulation of [24]. HLL approximate Riemann solver. Flux-CT scheme for divergence-free constraint. Cell reconstruction schemes: minmod, MC, CENO, and PPM. 2nd and 3rd-order TVD Runge-Kutta schemes for time update.
ECHO [91]	2D. HLL approximate Riemann solver. High-order reconstruction methods: minmod, MC, and ENO-like schemes (up to 5th-order accurate for smooth flows). Upwind constrained transport for divergence-free constraint. Runge-Kutta time stepping. Able to handle the limiting case of magnetodynamics.
	Artificial Viscosity
DVH03 [86]	3D. Nonconservative scheme. Boyer-Lindquist coordinates (Kerr black-hole spacetime). Staggered grid, time-explicit, operator-split finite difference scheme. Method of characteristics CT scheme for divergence-free constraint.
cosmos++ [20]	3D. Internal energy formulation and dual energy/flux-conserving formulation. Latter accurate for ultrarelativistic flows.