The superscripts (n+l,Jl) and (n+l,£+l) in Equations (23) and (24) refer to the 

 Jl-th and (Jl+l)-st corrected solutions of the system of differential equations 

 at time step n+1 . Generally, the stopping criteria for the iterative 

 procedure in the Euler-modif ied method at each time step are that the maximum 

 absolute change in the x- and y-values on the free surface and the body is 

 ej = 0.001 and the maximum relative change in the ({i-values on the free 

 surface is £2 = 0.001 for ((j-values greater in absolute value than 

 £3 = 0.000001. 



Each of the five discrete evolution equations has (j)x or <f)y on the right 

 side. In particular, to compute the right sides of Equations (21a-e), ^.^ and 

 <t>y along the moving boundaries of the fluid region must be computed from the 

 solution of the Laplace equation for ()>. The solution method for this 

 equation is described in the next section. 



To prevent numerical instabilities on the computed free surface from 

 arising, a linear filtering scheme due to Shapiro [12] is used. The filtering 

 scheme has been used successfully by Ohring and Telste [13], Haussling and 

 Coleman [14], and several other researchers. It has been applied at fixed 

 intervals of time, and has been especially helpful near the intersection of 

 the free surface with the body contour. 



Unless some method is used to maintain a reasonable grid spacing along 

 the free surface and the body contour, grid points will congregate in some 

 areas and become sparse in other areas. A method is used to keep a uniform 

 distribution of points along the body contour and a prescribed distribution of 

 grid points along the free surface. The prescribed free-surface distribution 

 is such that the free-surface length between the body and the j-th free- 

 surface grid point is a constant fraction of the total free-surface length 



13 



