Lagrangian Solutions of Unsteady Free Surface Flows 



B. Stability of Cell Relaxation 



Suppose the central member, cell A, of the group of cells 

 shown in Fig, 3 contained a residual R^ which was then dissipated 

 according to the relations (23). Transfer functions, D^^g, D^^^., etc., 

 will describe the residual changes, ^Rg» etc, in the surrounding 

 cells where 



AR3=coD^^R^, etc. 



(24) 



Fig. 3. Z-Plane 



For example 



D^B= {(Z, - Z,5)(Z2- Z^) - (Z,6 - Z^XZ, - 7^)}i/^A^ 



where A^ is the area of cell A. For convergence of the relaxation 

 method it is clearly necessary that the co for each cell be chosen 

 so that all Ci)|d| are significantly less than unity. It is instructive 

 to inspect the case in which all the cells are roughly geometrically 

 similar in the Z plane. Then 



D 



ABl 



= IDadI = ID 



AFl 



= D 



AHl 



8A, 



'2l _ 



129 



