  A continuum approach to study magnetoelectric multiferroic $\mathrm{BiFeO}_3$
(BFO) is proposed. Our modeling effort marries the ferroelectric (FE) phase
field method and micromagnetic simulations in order to describe the entire
multiferroic order parameter sector (polarization, oxygen antiphase tilts,
strain, and magnetism) self-consistently on the same time and length scale. In
this paper, we discuss our choice of ferroelectric and magnetic energy terms
and demonstrate benchmarks against known behavior. We parameterize the lowest
order couplings of the structural distortions against previous predictions from
density functional theory calculations giving access to simulations of the FE
domain wall (DW) topology. This allows us to estimate the energetic hierarchy
and thicknesses of the numerous structural DWs. We then extend the model to the
canted antiferromagnetic order and demonstrate how the ferroelectric domain
boundaries influence the resulting magnetic DWs. We also highlight some
capabilities of this model by providing two examples relevant for applications.
We demonstrate spin wave transmission through the multiferroic domain
boundaries which identify rectification in qualitative agreement with recent
experimental observations. As a second example of application, we model
fully-dynamical magnetoelectric switching, where we find a sensitivity on the
Gilbert damping with respect to switching pathways. We envision that this
modeling effort will set the basis for further work on properties of arbitrary
3D nanostructures of BFO (and related multiferroics) at the mesoscale.
