Abstract: The solid-liquid phase change process in the presence of the magnetic field has extensive and important applications in industrial engineering, i.e. the electromagnetic metallurgy and additive manufacturing, where the melting process and flow mechanisms have not been fully explored. The cavity melting model, as a basic problem, has good universality to study the solid-liquid phase change process, and it can also provide some basic information for the magnetohydrodynamics influence on the phase change problems. In this paper, the solid-liquid phase change process is simulated based on the enthalpy method, and the physical model of the cavity heated from the left wall is considered, with a transverse magnetic field perpendicular to the main circulation direction. The flow field, heat transfer and melting processes are investigated, focusing to the influences of different factors. At first, for the melting problem of a square cavity without a magnetic field, we compared our numerical solution with the experimental results reported by some other literatures, and it is confirmed that the influence of the cavity width on the profile and position of the solid-liquid interface cannot be ignored. Subsequently, we use the 3D model to simulate the cases in the presence of small magnetic fields, and it is found that the Lorentz force mainly acts as a rectification effect on the chaotic 3D flow, making the flow tend to be quasi-2D (Q2D). Meanwhile owing to the existence of the solid-liquid interface, the velocity field in the mainstream region tends to be more uniform under the external magnetic field, and thus the shape of the solid-liquid interface also transforms into the 2D structure correspondingly. Finally, the Q2D model is used to study the cases under greater magnetic fields, and the influences of different parameters on the heat transfer efficiency and interface shapes are discussed. Moreover, the scaling law to describe the relation between the max vertical velocity and other dimensionless parameters is also proposed, in order to quantitatively characterize the melting process under the transverse magnetic field.