supernova dynamics and thus are frequently neglected. Since strong gravity and velocities up to a few tenths of the speed of light are involved, the MHD equations are best solved in a general-relativistic formulation. General-relativistic MHD is computationally particularly expensive, because the conserved variables are not the primitive variables (density, internal energy / temperature, velocity, chemical composition). The latter are needed for the EOS and enter flux terms. After each update, they must be recovered from the conserved variables via multi-dimensional root finding.

-   **Gravity**. Deviations in the strength of the gravitational acceleration between Newtonian and general-relativistic gravity are small in the precollapse core, but become of order $10-20\%$ in the protoneutron star phase. In the case of black hole formation, Newtonian physics breaks down completely. General relativistic gravity is included at varying levels in simulations. Some neglect it completely and solve the linear elliptic Newtonian Poisson equation to compute the gravitational potential. This is done using direct multigrid methods or integral multipole expansion methods. Some codes modify the monopole term in the latter approach to approximate general relativistic effects.

    Including full general relativity is more challenging, in particular in 2D and 3D, since there general relativity has radiative degrees 