Computation of the Motion of Long Water Waves 



|?=v-u|3 (51) 



Many difference schemes may be developed to approximate Eq. (51). 

 Our tests show that the forward implicit method with the difference 

 equation 



n+l n < n+l n-I \ 



is one of the best. A stability analysis shows that Eq, (52) leads to 

 a linearly stable computation with slight dampling. 



Numerical tests were carried out in the context of a simple 

 physical problem whose exact solution was known, viz. , a solitary 

 wave in a horizontal channel. Among five alternative combinations 

 of surface and correction term treatments tested, that using Eq. (52) 

 and Eqs. (47) and (48) was the best. 



Now consider 6t. The maximum fluid speed in the above 

 tests was u^^^ « 0.30 and 6x = 0.5 while 6y = O.i. According to 

 Eq. (49) 



6t<^ = ^=1.67 



The speed of the surface wave is C = 1.18. The Courant condition 

 [ Eq. (50)] would require 



C^^SX 0.50 n ^-, y, 



^^<-C =TT8 =^-^^^ 



But the Courant condition should also be observed in computing 

 the free surface. Because we used the spacing A = 0.05 at the free 

 surface, the condition 



must be satisfied. Therefore, the most restrictive condition is 

 6t < 0.0424. In all the test examples, 6t = 0.05 was used. This is 

 slightly larger than the estimated maximum allowable 6t, but no 

 distortions or instabilities were noted. However, the result of 

 seriously violating the Courant condition, i.e., using 6t = 0.10, 

 was large non-physical distortions that suggest one has to be careful 

 about the choice of 6t. 



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