The purpose of this paper was to demonstrate the application of a procedure to predict internal sloshing loads on partially filled tank walls of liquefied natural gas (LNG) tankers that are subject to the action of sea waves. The method is numerical. We used a moving grid approach and a finite-volume solution method designed to allow for arbitrary ship motions. An interface-capturing scheme that accounts for overturning and breaking waves computed the motion of liquid inside the tanks. The method suppressed numerical mixing. Mixing effects close to the interface were buried in the numerical treatment of the interface. This interface, which was at least one cell wide, amounted to about 20–50 cm at full scale. Droplets and bubbles smaller than mesh size were not resolved. Tank walls were considered rigid. The results are first presented for an LNG tank whose motion was prescribed in accordance with planned laboratory experiments. Both two-dimensional and three-dimensional simulations were performed. The aim was to demonstrate that (1) realistic loads can be predicted using grids of moderate fineness, (2) the numerical method accurately resolves the free surface even when severe fragmentation occurs, and (3) long-term simulations over many oscillation periods are possible without numerical mixing of liquid and gas. The coupled simulation of a sea-going full-sized LNG tanker with partially filled tanks demonstrated the plausibility of this approach. Comparative experimental data were unavailable for validation; however, results were plausible and encouraged further validation.