T = 8 s, DX = 50 m, and D = 6 m. In the comparison, the time step, DT, was 

 varied and the reference or standard case was taken to be the explicit scheme 

 with DT = 6 hr. The relative accuracy with respect to the reference result, 

 Ay , where Ay is the change in shoreline position between final and initial 

 positions, is given at three locations on the left side of the beach. 



Table 1 



Stability and Accuracy of Explicit and Implicit 



Numerical Schemes with an Operative Seawall 





Stabili 



ty 









Ay - 



^r (percent) 





At 



parameter 

 R s 



Relative 

 Execution Time 



Explicit 



Scheme 





Ay r 







hr 



i = 1 





i = 10 





i = 20 











1 



0.08 





5.30 





-0.6 





-3.1 





0.0 



2 



0.17 





2.72 





-0.5 





-2.8 





0.0 



4 



0.34 





1.43 





-0.2 





-1.2 





0.0 



6 



0.51 





1.00 





0.0 





0.0 





0.0 



8 



0.67 





unstable 

 Implicit 



Scheme 













6 



0.51 





2.24 





-0.5 





-3.1 





0.0 



12 



1.01 





1.19 





-0.2 





-2.0 





0.0 



24 



2.02 





0.67 





-0.7 





0.0 





0.0 



60 



5.05 





0.35 





13.4 





2.3 





14.7 



120 



10.11 





0.24 





22.1 





-7.9 





23.3 



87. The results in Table 1 are qualitatively similar to those given by 

 Kraus and Harikai (1983). The explicit model is computationally faster than 

 the implicit model per time step; however, larger time steps can be taken with 

 the implicit model while preserving reasonable numerical accuracy, allowing a 

 potential overall speed advantage. For example, the implicit model with a 

 time step of 24 hr and stability parameter of 2.02 is about 30 percent faster 

 than the reference explicit result, yet still has acceptable numerical accu- 

 racy. Engineering judgment must be exercised on a case-by-case basis to de- 

 cide if a 24-hr time step will give acceptable physical accuracy. In a simi- 

 lar comparison without a seawall, Kraus and Harikai (1983) found the implicit 

 model with a 6-hr time step to be comparable in accuracy and execution time to 

 the reference explicit model with the same time step. As was discussed in 

 Part III, the implicit model suffers a loss in efficiency when the seawall 

 boundary condition is operative. 



37 



