The bulk replacement current density triggered by surface charge loss owing to thermionic emission leads to a volumetric Lorentz force which has been observed to drive macroscopic melt layer motion in transient tungsten melting tokamak experiments in which components of different geometries (deliberate leading edges and sloped surfaces) have been exposed to edge localized mode (ELM) pulsed heat loads in high power H-mode discharges. A self-consistent approach is formulated for the replacement current which is based on the magnetostatic limit of the resistive thermoelectric magnetohydrodynamic description of the liquid metal and results in a well-defined boundary value problem for the whole conductor. A new module is incorporated into the incompressible fluid dynamics code MEMOS-3D, which numerically solves the finite difference representation of the problem. The phenomenological approach, employed thus far to describe the replacement current, is demonstrated to be accurate for the sloped geometry but inadequate for the leading edge. MEMOS-3D simulations of very recent ASDEX-Upgrade leading edge experiments with the rigorous as well as the simplified approach are reported. For these simulations, the self-consistent approach predicts a fivefold reduction of the displaced material volume, a sevenfold reduction of the maximum peak height of displaced material and a different eroded surface morphology in comparison with the previously applied simplified approach.