  We investigate the multi-domain structures in the tetragonal and orthorhombic
phases of BaTiO$_3$ and the impact of the presence of domain walls on the
intermediary phase transition. We focus on the change in the transition
temperatures resulting from various types of domain walls and their coupling
with an external electric field. We employ molecular dynamics simulations of an
ab initio effective Hamiltonian in this study. After confirming that this model
is applicable to multi-domain configurations, we show that the phase transition
temperatures strongly depend on the presence of domains walls. Notably we show
that elastic 90$^{\circ}$ walls can strongly reduce thermal hysteresis. Further
analysis shows that the change in transition temperatures can be attributed to
two main factors - long-range monoclinic distortions induced by walls within
domains and domain wall widths. We also show that the coupling with the field
further facilitates the reduction of thermal hysteresis for orthorhombic
90$^{\circ}$ walls making this configuration attractive for future
applications.
