street^ Chan and Fromm 



P values replaced by the n+ 1 values just calculated, the accuracy 

 of the solution is increased; this is the corrector iteration. Numeri- 

 cal tests showed that the computations remained stable if at least 

 two iterations were used (one predictor, one corrector). The u"* , 

 v"*' and T|"*' values obtained in the second and the third iteration 

 agreed to at least four significant figures after several hundred At 

 steps in simulation of solitary wave motion onto a shelf (Fig. la). 



Boundary conditions in difference form were derived from 

 Eqs. (7-10) in the case of solitary wave simulation. The wave was 

 started well inside the tank walls which were held rigid. For ex- 

 ample , for Wall 1 in Fig. 1 we have from Eqs. (7-10) at any time 

 level "Hij = "Hsi* V|j = V3J, U2i= 0, and U|. = - u^:. Other walls have 

 similar conditions. 



For input of an oscillatory wave propagating in the positive 

 x-directlon at x = we prescribe 



TiS;' =Ti^sin[-^- At. (n+1)] (22) 



' -"-0 



where r\Q is the amplitude (usually small) and Lq is the wave 

 length. The celerity Cq is taken to be unity in nondimensional 

 terms. From Eq, (14) we have v,: = Vjj, Because r|^t' and ti^j"' 

 are computed explicitly in the tank region, r]^'^^ is obtained by 

 polynomiial interpolation according to the second-order formula 



Finally, the continuity difference Eq, (21) is used for points (2j) 

 where it has not been previously employed to relate uj'p to the 

 values in the interior, j ^ 2. With u,. known as a function of 

 Ugj, U3:, etc, , Eq. (19) can be used in 2 ^ i ^ M-2 and the u"* 

 found; v9.*' values are replaced by vfj values in the first iteration. 



Figure 2 is a flow chart for the APPSIM2 computations. 

 These were performed on an IBM 360/91 system. For a typical 

 computation with A = 0. 25 , At = 0. 22 , M = 154, N = 54 and 126 

 time steps the program required about 360K bytes (90K words) of 

 core storage and 4 minutes of CPU time (about l/30 minute per 

 time step). The stability of the method is discussed in Sec. 3.4 

 after presentation of computational results. 



3.3. Results and Discussion 



To illustrate the focusing effect of wave refraction and the 

 reaction of waves to a shelf geometry, solitary and oscillatory waves 

 were shoaled over the bottom topography shown in Fig. la. The 

 water depth in the tank was 1.0 while the depth of the shelf was 0.4. 



160 



