On the evaluation of time-domain Green function

A note on the relative efficiency of methods for computing the transient free-surface Green function

“…The solution to Eq. (9) developed by Chuang et al (2007) is based on a truncated Taylor series expansion of F (µ, τ ) (at a fixed value of µ) about the starting time τ 0…”

“…17into the ODE and collecting terms at each power of τ , recursive formulae can be derived for the coefficients a n based on the initial values of Y . These relations are given by Equations (26) and (27) of Chuang et al (2007) but it is important to note that both of these equations are missing a minus sign on the right hand side. The computational effort required to evaluate the coefficients a n , and then compute the solution at the next time-step is proportional to…”

“…They also imply that the method is more efficient than the Taylor expansion method of Chuang et al (2007), though this is not demonstrated. Comparing two numerical methods at the same time-step size is however only relevant when the two methods require the same computational effort per time-step, which is not the case here.…”

“…Clement adopted the classical explicit four-step, fourth-order Runge-Kutta scheme (RK44) to integrate the ODE and compute the Green function, as did several other subsequent authors including Duan and Dai (2001); Liang et al (2007). Ten years later, Chuang et al (2007) developed a closed-form solution to the ODE based on Taylor series expansions, allowing the Green function to be computed by truncating the series at a finite number of terms. Although not explicitly stated, the implicit intent of this work was to provide a more efficient means of solving the ODE compared to the RK44 method, even though no demonstration of that result was provided.…”

Use of Clement’s ODEs for the Speedup of Computation of the Green Function and its Derivatives for Floating or Submerged Bodies in Deep Water

“…Much later [1], he transposed those fourth order time domain ODE into second order ODE for the free surface Green function in the frequency domain. For the time-domain Green function, a semi-analytical precise integration method [11] and a analytical method based on Taylor series expansion [12] have been developed replacing the standard fourstep Runge-Kutta method (RK4) to solve the fourth order ODEs. Recently, still in time-domain, Bingham [13] compared the efficiency of three ODE-based methods and the standard algorithms developed by Newman [14].…”