A precise computation method of transient free surface Green function

The transient free surface Green function (TFSGF) is of great importance in the prediction of unsteady ship motions with forward speed. In this numerical approach, the wave part of the TFSGF and its spatial derivatives are obtained by solving the fourth-order ordinary differential equations (ODEs) based on the semi-analytical Precise Integration Method (PIM). Theoretical derivations show that in fact the horizontal and vertical derivatives can be expressed by TFSGF, which means only one ODE needs to be solved. The stability and accuracy of this method is demonstrated by the comparison with other method as well as the analytical solutions. Additionally, the proposed method is applied to solve the radiation problem of a floating hemisphere at zero speed using the time domain Rankine–Green method. The numerical hydrodynamic coefficients show good agreement with the analytical solutions. •The three ordinary differential equations of the Green function are simplified to one.•The ordinary differential equation is solved by the Precise Integration Method.•The radiation problem of a hemisphere is solved by the time domain Rankine–Green method.•The proposed method can compute the Green function accurately, efficiently and stably.