Headcut erosion is a special phenomenon of riverbed erosion and deposition, involving transcritical flows and severe riverbed deformation processes. In the numerical simulation of the erosion process, the numerical oscillation caused by the alternating of torrent and slow flow often occurs. The main reason for this problem is that the Preissmann scheme is only suitable for simulating continuous water flows without interruption points, it cannot simulate transcritical flow because of the occurrence of interruption points. According to the Froude number of each cross-section, the position of each interruption point was identified, and the channel was divided into different flow sections to ensure that each flow section had a continuous flow with the same flow pattern. Appropriate boundary conditions were proposed for the four cases of slow flow, rapid flow, hydraulic jump and water fall, respectively. Then a one-dimensional mathematical model of unsteady riverbed mass transport based on Preissmann scheme for transcritical flow headcut erosion was established, which was verified afterwards using flume test data. The results show that the predicted value is in good agreement with the measured value, demonstrating the reliability and applicability of the model.