The Numerical Algorithm Of Ordinary Differential Equation For Frequency Domain Green Function

The Green function of free surface infinite depth case oscillatory source is very important.It is important tool to calculate added mass,wave damping coefficient,wave force and wave moment and has great significance in practice.This paper focuses on the numerical algorithm of frequency Green function.Our jumping-off place is second ordinary differential equation for frequency Green function and different from many published papers.The following is our main research content:Firstly,the second order differential equation frequency Green function published by Clement in 2013 is given.Derive gradient Green functions G_r,G_Zdifferential forms.Next,derive the integral term F（X,Y）of Green function and its first and second order derivatives formulas.The rezoning issues of calculation is explored.The domain is divided into five sub-areas.In each subarea,use series expansion correspond to its property and study the maximum terms of series expansion within requirement precision.Then,the efficient algorithm is given to the Green function G and its gradient functions G_r,G_Zin two regions（ω≥1,ω<1）.As to theω≥1 region,the TBNM method is introduced.According to the linear of Green function and its gradient functions,the special TBNM method is derived to them and the representation is matrix form.Compared with classical Runge-Kutta method,the computing speed of TBNM method is twice faster than it and TBNM mehtod has six-order accurate that is higher than the four-order accurate of Runge-Kutta.As to theω<1 region,the power series solution formulas is derived.The recurrence relation to compute coefficient of power series is given.Finally,the value of integral terms F（X,Y）and its first order and second order derivatives compare with standard value by Newman computed.Tested in desktop with 4G memories,the new al-gorithm calculation precision reach 10^-9and calculation time within 0.001s to each node.Give the relation of F（X,Y）second order derivative between calculation value and series terms.The experiment result show that the TBNM algorithm calculation precision reach 10^-6and the power series method reach 10^-7which can be higher.In terms of computation speed,both TBNM algorithm and power series method are faster than compute Cauchy integral.The ultimate experiment results show that,to the high requirement of calculating preci-sion and calculating efficiency for the oscillating second ordinary differential equation of Green function,the combine TBNM algorithm with power series method is a highly efficiency com-bined method.