Computation of the Motion of Long Water Waves 



The evolution of fluid dynamics is calculated in "cycles," or 

 time steps. At the start of each cycle the source term Rjj for 

 each cell is evaluated by Eq. (43). The pressure p is computed 

 only for those cells whose centers fall in the fluid region; either 

 Eq, (41) or Eq. (44) is used as appropriate. The successive-over- 

 relaxation method is used to solve the p field. The iteration is 

 terminated when 



(m) (m-l) 



< Cp (46) 



for every cell, where (m) means the m iteration and e is a 

 predetermined small positive number. The accuracy in solving pij 

 at the n time step has a direct bearing on the accuracy of satisfy- 

 ing the continuity equation D"*' = [Eq. (4i)] at the n+l^** step. 

 Smaller values of Dj"j result when smaller Cp are used. However, 

 there is little improvement in reducing Djj for Cp < 10' because 

 the round-off level of the computer has been reached. 



Now Eqs. (37) yield the new velocities. Then each marker 

 particle is advanced to its new position by Eqs. (45). Thus a cycle 

 is completed and the next one can be started immediately. 



The convective contributions given by Eqs. (38) and (39) can 

 be approximated by a wide variety of finite difference formulas. 

 Chan and Street [ 1970c] show that, while the original MAC and 

 early SUMMAC equations used a first-order explicit method, second- 

 order explicit methods are better. Of the two second-order explicit 

 schemes studied, the so-called "upstream" difference alone rather 

 than in a "phase-averaged" procedure yields better results in prob- 

 lems where free surface waves are present. In this upstream dif- 



ference, if w^^ represents either Uj^ij or Vji^i , 

 case when uj > and -v^^ > 



then for the 



* 



■"He ^3-1 



Im im-1 2 Im-Z ir 



:m 



.2 



+ ^(-i™..e-2»i„., +»iJ (^^) 



where 



w« = w, 



n , or- 1 , n n 



't - -^ a i™ + o (w. _„ - w.^) 

 im i-lm 2 i-2m im' 



^ ^ (»;-.» -^-i"-,.^-; J <*«) 



and 



175 



