An efficient algorithm with new residual functions for the transient free-surface green function in infinite depth

Efficient evaluation of the transient free-surface Green function with sufficient accuracy is the key to employ time-domain panel method in solving hydrodynamic problems. In this paper, series and asymptotic expansions for the Green function and its two spatial derivatives are deduced in detail for the evaluation. New residual functions are derived straightforwardly from the resulted asymptotic formulas and it is concluded that the present residual functions decay rapidly than existing choices. The ultimate algorithm is the combination of Chebyshev approximations and asymptotic expressions, along with a special partitioning line introduced to speed up the evaluation. After comprehensive numerical validation, it is pointed out that the present algorithm owns at least 6 digits of accuracy and sufficient efficiency for practical application. Any efficient method for evaluating the Green function will use the asymptotic expansions. Furthermore, it should be pointed out that the present residual functions can also be conveniently applied to the scheme of interpolation in a precomputed table.