APPENDIX E 



Advective Schemes 



The finite-difference simulation of dispersion of contaminants (heat, 

 salt, sediments, etc.) over large coastal regions is often carried out with a 

 limited number of grid points. In this case, the choice of the convective 

 scheme becomes very cruci-al. In general, it is desirable to have a convective 

 scheme which can maintain accuracy, positivity, and conservation in the 

 results. 



In this appendix, four convective schemes with differing amounts of 

 numerical diffusion and dispersion are compared. These schemes are: (1) the 

 upwind difference scheme (Roache, 1972), (2) the combined upwind and central 

 difference scheme (Sheng, 1975), (3) the central difference scheme with 

 smoothing (Sheng, Segur, and Lewellen, 1978), and (4) the flux-corrected 

 transport scheme (Boris and Book, 1976; Zalesak, 1979). The details of these 

 schemes are briefly described in the following. 



We will consider the conservation equation of a passive contaminant in 

 two spatial dimensions (x and z): 



i^.^.^=0 (E-l) 



3t 3x 3z 



where the flow field (u,w) is computed from some hydrodynamic model. Various 

 numerical schemes have been developed to approximate the convection terms in 

 the above equation. Numerical schemes introduce errors in the form of 

 artificial numerical diffusion and/or numerical dispersion which often 

 manifests itself in the form of short-wave oscillations (with wavelength 

 typically twice the spatial grid spacing). In the present section, four 

 convective schemes are described with respect to Eq. (E-l) and the numerical 

 grid shown in Figure E.l. The finite difference form of Eq. (E-l) can be 

 written in the following general form: 



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