between the body and the outer boundary. The scheme allows one to follow the 

 movement of boundary grid points within a time step as if they were fluid 

 particles and to shift the grid points to other fluid particles at the end of 

 each time step. To shift the grid points, cubic spline interpolation is used 

 to fit the arclength as a function of grid point number along the free surface 

 and the body contour. The positions of the grid points along the free surface 

 and the hull are shifted and the values of all pertinent functions 

 interpolated, using the cubic splines, to their new values. The 

 redistribution of grid points, of course, affects the initial guess for the 

 Euler-modif led method at time step n+I. However, if the shifting is done 

 every time step and the time step is sufficiently small, the redistribution 

 scheme has been found to proceed smoothly. 



Because of numerical errors, grid points cannot be expected to remain 

 exactly on the body as the solution of the initial -boundary value problem is 

 advanced in time. Numerical errors arise from the redistribution scheme and 

 from replacing the differential equations by finite difference equations. To 

 correct for such errors, grid points that move off the body are shifted 

 back to the body. This is accomplished by computing the counterclockwise 

 angle about the center of the body from the direction of the positive x-axis 

 (Fig. 2). Every body grid point at a location slightly off the body surface 

 at a certain value of that angle is relocated to the point on the body having 

 the same value of that angle. 



At the Intersection of the free surface and the body contour difficulties 

 arise. At this point both the free-surface boundary conditions and the 

 boundary conditions associated with the solid boundary apply at a particular 

 instant of time. However, the fluid particle at the intersection may move to 



14 



