Pressure and velocity are generally discontinuous at the solidification front when the solid domain is deformed. This discontinuity, which has been ignored in most conventional solidification models for continuous casting processes, could cause significant numerical error around the solidification front in numerical simulations. To overcome error, a solidification front-tracking model in which the liquid-solid interface is discontinuous for pressure and velocity was developed. Assuming a sharp solidification front at the solidification front, the liquid and the solid domains are weakly coupled for momentum transport equations. The present model was proved to be more effective in cases of higher solidification rate such as in twin roll rapid cooling casting.