y- sweep 







1 



2At 



, k+1 

 (n 



- n*) + — - 

 2Ay 



1 



2At 



, k+1 

 (v 





S (v'^+^d - v'^-^d) = (16) 



y 



,. S (n*"^^ + ri^'b = (17^ 



2Ay y 



14. The details of applying the SC scheme to Equations 1-3 can be found 

 in a report by Butler (in preparation). The diffusion terms of Equations 2 

 and 3 are also represented with time-centered approximations. The inclusion 

 of diffusion terms contributes to the numerical stability of the scheme 

 (Vreugdenhill 1973), and serves to somewhat account for turbulent momentum 

 dissipation at the larger scales. While the resulting finite difference forms 

 of Equations 1-3 appear cumbersome, they are efficient to solve. Application 

 of the appropriate equation to one row or column of the grid (the "sweeping" 

 process) results in a system of linear algebraic equations whose coefficient 

 matrix is tridiagonal. Tridiagonal matrix problems can be solved directly, 

 without the cost and effort of matrix inversion. 



15. The computational time-step for the SC scheme in WIFM js largely 

 governed by t^imple mass and momentum conservation principles. The maximum 

 time-step for a problem is characterized by: 



"■ At ^ ^ (18) 



where V is the largest flow velocity to be encountered at a cell with its 

 smallest side length AS . The parameter n is of order 1. So, the time 

 step is constrained by the smallest cell width which contains the highest flow 

 velocity. In physical terms. Equation 18 requires that the flow cannot move 

 substantially farther than one tell width in one time-step. 



Bo undary Conditions 



16. WIFM allows a variety of boundary conditions to be specified, which 

 can be classified into three groups: open boundaries, land-water boundaries, 

 and thin-wall barriers. • 



Open boundaries 



17. When the edge of the computational grid is defined as water, such 



16 



•^>ii3t*'UliMnitKt^''^ 



