  We have performed an extensive test of the ability of density functional
theory within several approximations for the exchange-correlation functional,
local density approximation+Hubbard $U$ and local density approximation +
dynamic mean field theory to describe magnetic and electronic properties of
SrRuO$_3$. We focus on the ferromagnetic phase, illustrating differences
between the orthorhombic low temperature structure vs the cubic high
temperature structure. We assess how magnetism, spectral function, and cohesive
properties are affected by methodology, on-site Hubbard $U$ and double counting
corrections. Further, we compare the impact of the impurity solver on the
quasiparticle weight $Z$, which is in turn compared to experimental results.
The spectral functions resulting from the different treatments are also
compared to experimental data. The impact of spin-orbit coupling is also
studied, allowing us to determine the orbital moments. In the orthorhombic
phase the orbital moments are found to be tilted with respect to the spin
moments, emphasising the importance of taking into account the distortion of
the oxygen octahedra.
