Computation of the Motion of Long Water Waves 



and 



-7 



ifx, I w 1 + 7 X 10 < 1 + o(q') 

 l^2,3Lax « 1.005 < 1 + 0(Qr) 



for \j > 2A. For X., = 2/^, |(x| = 1 exactly. Forsythe and Wasow 

 [ I960j suggest that the errors may be controllable and lead to a 

 stable computation when 



|ji| < 1 + 0{^t) (32) 



In the present case or < At because or = uAt/A and u/A < 1; 

 accordingly, the condition (32) can be written as ||i| ^ 1 + 0(a) . 

 Our computationeil experience with APPSIM which has 02 = always 

 and does not use iteration also suggests that the computation is stable, 

 at least for several hundred time steps. APPSIM2, however, because 

 of its coupled (u, v, T|) equations and the propagation of error from 

 the u-field where |ujj | is relatively large compared to the v- field 

 and T|-fleld, does require at least one iteration (which tends to make 

 the connputation more like an implicit scheme) to retain a smoothly 

 varying solution on a smoothly varying bottom topography. 



3.5. Prognosis 



The computational results indicate that APPSIM2 is a useful 

 means of studying the evolution of flow fields in wave shoaling over 

 smoothly varying bottom topography. However, the method requires 

 considerable computer storage and moderate execution time. Thus, 

 APPSIM2 , which models nonlinear processes in nonbreaking waves, 

 should be used only when nonlinear effects are expected to be sig- 

 nificant, other methods (cf. , Section I ) being appropriate otherwise. 

 As Madsen and Mei [ i969a] indicate, the equations of KdV type used 

 In APPSIM2 should make the method applicable to a wide range of 

 long wave problems. 



Two futher steps should be made in the development of the 

 method. First, the linear stability property could be improved by 

 introduction of a second-order central difference method for the 

 convective terms in Eqs . (18) and (21). This central difference 

 [Fromm, 1968] leads to a modification of Eq. (31) such that 



IM-Z.slmox 



< 1 



for most components of interest. Alternatively, Eq. (21) can be 

 made an entirely implicit equation for T|"j* [ cf . , Eq. (19)] ; this 

 will eliminate the growing contribution represented by Eq. (31) and 



169 



