Ultra-luminous X-ray sources (ULXs) are off-nuclear, compact, very luminous X-ray souces
with X-ray luminosity exceeding 10^{39} erg/s. There has been a long debate regarding
the central engine of ULXs: they could be super-critically accreting stellar-mass black
holes or sub-critically accreting intermediate-mass black holes because the ULX
luminosities exceed the Eddington limit for stellar-mass black holes. The recent
discoveries of periodic pulsed X-ray emissions in 3 ULXs (called ULX-pulsars) show that
at least a part of ULXs are powered by super-critically accreting neutron stars
(Bachetti et al. 2014, Israel et al. 2016). The discoveries of ULX-puslars are also
strong evidences that super-critical accretion can actually occur in the
universe. However, the dynamics and radiative spectral features of super-critically
accreting black holes and neutron stars are poorly understood. Especially, it is not
clear whether the super-critical column accretion  onto magnetized neutron stars can
occur. 

We, therefore, performed radiation hydrodynamic simulations of super-critical accretion
flows onto black holes and strongly magnetized neutron stars to approach these
problems. It was found that super-critically accreting stellar-mass black holes can
explain the luminosity and spectral features of ULXs if they do not show the pulsed
emissions. We also present that the super-critical column accretion onto strongly
magnetized neutron stars occurs and the luminosity of the accretion columns can reach
the ULX-pulsar luminosities. Our results support the super-critical accretion scenarios
for ULXs.