Extension of a multi-region free-surface MHD solver beyond the inductionless approximation

Free-surface liquid metal flows are a leading candidate for the plasma-facing components of future fusion reactors, but existing transient, three-dimensional, free-surface magnetohydrodynamics (MHD) solvers rely on the inductionless approximation, in which the induced magnetic field is neglected. This paper extends the open-source solver FreeMHD [Wynne et al., Phys. Plasmas 32, 013907 (2025)] beyond that approximation, resolving the induced field self-consistently with a vector potential formulation that enforces ∇·B=0 by construction while preserving the original multi-region, two-phase framework. It is verified against the analytical Shercliff and Hunt duct flows, against flows driven by a time-varying applied field, and against the deformation of a free liquid metal jet crossing a non-uniform field, and validated against free-surface height measurements from the liquid metal experiment-upgrade experiment. The experiment validates the overall free-surface solution rather than finite-Rm effects, which the transient-field cases verify. To our knowledge, this is the first open-source, fully three-dimensional free-surface liquid metal solver to resolve the evolution of the induced magnetic field, providing a basis for modeling the finite magnetic Reynolds number conditions expected in large-scale, transient fusion events.