  We have investigated the effect of epitaxial strain on the magnetic
properties and $B$-site cation ordering in multiferroic Bi$_2$FeCrO$_6$ (001)
thin films using a density-functional theory approach. We find that in thin
films with rock-salt ordering of Fe and Cr the ground state is characterised by
C-type anti-ferromagnetic (AFM) order. This is in contrast to the bulk form of
the material which was predicted to be a ferrimagnet with G-type AFM order.
Furthermore, the cation ordered thin-films undergo a transition with epitaxial
strain from C to A-type AFM order. Other magnetic orders appear as thermally
accessible excited states. We also find that $B$-site cation disordered
structures are more stable in coherent epitaxial strains thereby explaining the
lowered magnetic moments observed in these samples at room temperature. Strain
varies both the sign as well as strength of the Fe-Cr superexchange coupling
resulting in a very interesting phase diagram for Bi$_2$FeCrO$_6$ thin films.
