steps (e.g., Peregrine, 1967, Camfield and Street, 1969^^, Madsen and Mei, 

 1959'+9^ Chan and Street, 1970^°) must be reviewed with caution since the 

 numerical accuracy of the results is questionable. This is because the 

 truncation errors from the acceleration terms were of the same order or 

 lower order than the Boussinesq terms and essentially masked its effect. 

 These researchers were interested in the physical aspects of wave trans- 

 formations which were qualitatively simulated so that the value of these 

 early efforts from this aspect is not diminished. 



In addition, all numerical methods must' use only a discrete number of 

 points per wavelength (N) to define each wave. Finite-amplitude (cnoidal) 

 waves when decomposed by Fourier analysis consist of many, superimposed 

 sinusoidal components. All harmonic components are described by fewer and 

 fewer grid points, and this coarse description can lead to amplitude and 

 phase errors. Thus, solutions of the Boussinesq equations are extremely 

 sensitive to numerical errors (Abbott and Rodenhuis, 1972^^, Hauguel, 1980). 



a. Abbott, Petersen, and Skovgaard (1978a, 1978b) . These researchers 

 of the Danish Hydraulics Institute were the first to successfully develop a 

 two-dimensional numerical model to accurately propagate quasi-long waves 

 over variable bathjmietry to near the breaking limit. They integrated equa- 

 tions (144), (145), and (146) using a staggered, implicit finite-difference 

 scheme. One key aspect of their contribution (see also Abbott, 1978, 1979) 

 is the method to remove all truncation error terms of order comparable to 

 the Boussinesq term so the resulting finite-difference equations are third- 

 order accurate. This was done (following the observations by Long, 1964 ) 

 by rewriting all higher order truncation error terms using the linearized 

 wave equations with no loss of accuracy or generality. In one dimension 

 these linearized equations are 



^ + -|£ = (151) 



9t 8x 



1^ + gd 1^ = (152) 



3t ^ 9x 



In this way, the truncation errors are finite-differenced, combined with 

 the finite-difference Boussinesq term and subtracted out of the equations 

 together in an efficient manner. It is theoretically possible to use 

 higher order accurate finite-differences initially to accomplish this goal 

 but Abbott, Petersen, and Skovgaard (1978a) claim this leads to "...algo- 

 rithmically intractable difference forms of the Boussinesq equations." 



S^CAMFIELD, F.E., and STREET, R.L. , "Shoaling of Solitary Waves on Small 

 Slopes," Joia>nal of the Waterways and Harbors Division, Vol. 95, No. WWl, 

 Feb. 1969, pp. 1-22 (not in bibliography) . 



^°CHAN, R.K.C., and STREET, R.L. , "Shoaling of Finite-Amplitude Waves on 

 Plane Beaches," Proceedings of the 12th Coastal Engineering Conference, 

 American Society of Civil Engineers, 1970 (not in bibliography). 



^^LONG, R.R., "The Initial Value Problem for Long Waves of Finite Amplitude,' 

 Journal of fluid Mechanics, Vol. 20, Pt. 7, 1964, pp. 161-170 (not in 

 bibliography) . 



153 



