x(0) = 

 Fj 



e 



J 



y(0) = 



Fj 







<t,(o) = 



Fj 







for £ = j or k, m = n or n+1, and G = B or F. The initial conditions become 



(22a) 



(22b) 



(22c) 



x(0) = X (a(e ), 0) (22d) 



Bk B k 



y(0) = y (a(e ), 0) (22e) 



Bk B k 



In Equations (21a-e) and Equations (22a-e), the subscript j runs over all 

 possible free surface grid points, and the subscript k runs over all possible 

 body grid points. The intersection of the free surface and the body is 

 treated as a body point obeying Equations (21d,e) subject to an initial 

 condition given by Equations (22d,e). 



The numerical scheme is implicit. An initial estimate for the five 

 functions at the (n+l)-st time step is obtained by linearly extrapolating from 

 two previous time steps. (For the first time step, the initial estimate is 

 the initial condition.) The functions are corrected Iteratively by using 

 Equations (21a-e). The Iterative procedure is stopped when the x- and y- 

 values have satisfied an absolute error criterion of the form 



|f(n+l,£) _ f(n+l,^+l)| < ^^ (23) 



and the ())-values along the free surface have satisfied the relative error 

 criterion given by 



|1 _ ^(n+l.£)/^(n+l,il+l)| < ,^ whenever |<i,(n+l.^+l) | > ^3 



12 



