Comparison of Four Schemes 



The problem considered here is the transport of a passive dye by 

 wind-driven currents in a two-dimensional basin (Figure E.3). Initially, dye 

 of 1000 mg/£ concentration is injected into the four numerical grids on the 

 upper left corner while the background concentration is zero everywhere. The 

 concentration contours at the end of one-day simulation are shown in 

 Figures E.4 through E.7. Due to the strong numerical diffusion of the upwind 

 scheme, the peak in Figure E.4 has dropped to 454 and the front of the 

 100 mg/Si contour has almost reached the right boundary. The results of the 

 combined upwind and central difference scheme in Figure E.5 are appreciably 

 improved over the upwind results. The central difference scheme with 

 smoothing maintains the peak closest to its original value of 1000 mgA, as 

 shown in Figure E.6. The smoothing scheme generally works quite well in the 

 interior region (Sheng, Segur, and Lewellen, 1978), but is not able to remove 

 a few negative concentration values adjacent to the boundary. The 

 flux-corrected transport scheme, as shown in Figure E.7, is able to maintain a 

 peak close to that in Figure E.6, and in the mean time does not produce 

 negative concentration values. 



In conclusion, it appears that both the central difference scheme with 

 smoothing and the flux-corrected transport scheme are superior to the upwind 

 scheme. In the absence of negative concentration values, the two schemes are 

 quite comparable. If the problem of interest contains sharp gradients near 

 the boundaries, or contains very little background diffusion, the 

 flux-corrected transport scheme should be used. Otherwise, either scheme may 

 be applied. The ability of the smoothing scheme in controlling short-wave 

 oscillations has also been demonstrated in realistic meteorological problems 

 (Lewellen and Sheng, 1981). 



260 



