2 2 „ 

 A Y A < 

 * L 



(E-7) 



where 



^' = (Vj,i + Vj_i - 2Vj)/(Ax)' 



\ = (Vj - Vj.2 - 2Vj.i)/(Ax)' 



If Eq. (E-5) and either Eq. (E-6) or (E-7) is true, smoothing is applied in 

 the following manner: 



where V,- is the smoothed result at C, and B is a positive constant. 



(E-8) 



First, the central difference scheme in the following form is applied to 

 solve Eq. (E-1): 



'R 



^R 



• (Ci,i,k ^C,-.k)/2 



(E-9) 



At the end of a computational step, the solution profiles along the horizontal 

 direction are first checked and smoothing is applied when necessary. Profiles 

 along the vertical direction are then checked and smoothed in a similar 

 manner. The smoothing scheme performed remarkably in many of the problems we 

 tested (Sheng, Segur, and Lewellen, 1978; Lewellen and Sheng, 1981). 

 Extensive tests indicated that B = 1/4 and u = 4 gave the best results. 



Flux-Corrected-Transport (FCT) Scheme 



This scheme was developed originally by Boris and Book (1973), 

 subsequently improved several times (Boris and Book, 1976), and most recently 

 implemented by Zalesak (1979). This method is a two step method involving a 

 low order calculation and a correction to a higher order. The correction is 

 controlled to maintain stability and sharp change of the physical variables. 

 The upwind scheme is used to compute the first order result: 



258 



