street^ Chan and Fromm 



6x ' ^ 6y 



We examined the finite difference convection equations by the 

 extended von Neumann method in which nonlinear equations are first 

 linearized. The resulting criteria were 



kl^l; ||3|:^l (49) 



A second criterion 



where C = the surface wave celerity, was derived by considering 

 the propagation of the free surface waves. These were simple linear 

 analyses and can only be used as guidelines in choosing the time 

 increment 6t for given 6x and 6y. Because numerical dispersion 

 is quite severe for short wave components, care must be exercised 

 to provide adequate resolution for all the important features in the 

 flow. As a rule of thumb, the smallest significant flow feature must 

 be represented by at least ten cells. 



In both the MAC and SUMMAC a line of particles was used to 

 mark the free surface position. A pair of (xj|, yj^*) values were 

 associated with the k*** particle at the n*** time step. Then Eq. (45) 

 was employed to calculate (xj*', yj*')- This procedure is really a 

 Lagrangian method that tends to be unstable after a large number of 

 time steps. The problem is not serious for simulating solitary 

 waves [cf. , Chan and Street, 1970a] . But, in calculating periodic 

 waves a given particle is moved up and down as each wave passes. 

 In the process a small number is systematically added to and then 

 subtracted fromi X|, and y^^ contributing to very large round- off 

 errors. In addition, there is no restraint on the individual particle 

 positions because each is calculated independently of the others. 



To overcome the difficulty with moving particles, an alterna- 

 tive approach using the Eulerian point of view can be developed. The 

 flow region is divided by a nunnber of vertical lines with equal spacing 

 ^ and T| is now the height of the free surface measured from the 

 reference level y = at the channel bottom. The horizontal posi- 

 tions of these vertical lines are fixed and we only compute the change 

 in r\ along each vertical line as time passes. 



The kinematic condition at the free surface, from the Eulerian 

 viewpoint is 



176 



