A time-domain numerical model with the linear free surface condition and large amplitude body motions is developed based on the higher-order boundary element method. Five numerical formulations are used to compute the nonlinear radiation forces on a moving body in the domain. A series of submerged bodies that have sharp edges and rounded corners with small radii are considered to investigate their reliabilities. It is found that for the bodies with rounded corners the nonlinear radiation forces calculated by the five numerical formulations are close to each other, while large discrepancies exist for the body with sharp edges. Examination on the vertical nonlinear radiation forces due to the body surge motion reveals that the body edge has less influence on the method of the rate of change of the fluid momentum; however, it has a stronger effect on the other four formulations. The former applied a substitution of the spatial derivatives of the velocity potential, and the other four use a direct integration of the fluid pressure on the body surface.