Compact binary systems involving both neutron stars and black holes with magnetized accretion disks are key sources for present and future multi- messenger astronomy observations. Theory and computational astrophysics models can help us to gain important insights of the complex plasma physics and astrophysics environments associated with these sources. For over a decade, we have been engaged in a program to perform long-term, accurate, GRMHD simulations of the gaseous environments close to merging supermassive black hole binaries. Our choice has been to adopt multiple computational strategies designed at accurately resolving both the long- term accretion dynamics of the resulting circumbinary disk and the highly relativistic flow of magnetized matter around each of the black holes.  Using this strategy, we recently performed simulations of accreting inspiralling spinning black holes that show promising hints of powerful jet launching and enhanced characteristic light signals that observers should search for when looking at the center of galaxies. Lastly, I will describe a similar computational framework developed in the context of our theory and computation astrophysics network (TCAN) that can be applied to perform very long-term simulations of binary neutron star post-mergers. I will show preliminary results of such simulations when realistic EOS and neutrino treatments are included.