x_(i,j,n) - (K Q(i,j,n-l)/D 2 )Q Q *(i,;j,n+l) (F-l) 



and 



APPENDIX F 



NUMERICAL ANALOGS OF SURGE EQUATIONS 



Admissible finite difference analogs of equations (40) and (41) 

 which are nominally centered at i , j , n (bearing in mind the 

 storage of variables in Figure 36), are as follows: 



[Q s *(i,j,n + 1) - Q s *(i,j,n-l]/2At 



- f[Q T Ji,j,n+l) + Q T *(i,j,n-l)]/2 



+ (gD/F(i,j)u(i))[H(i+l,j,n) - HgCi+l.j ,n) 



- H(i-l,j,n) + H B (i-l,j,n)]/2AS* 



[Q T ,(i,j,n + l) - Q T .(i,j,n-l)]/2At 

 + f[Q s Ji,j,n + l) + Q s ^(i,j,n-l)]/2 

 + (gD/F(i,j)v(j))[H(i,j + l,n) - H B (i,j + l,n) 



- H(i,j-l,n) + H B (i,j-l,n)]/2AT* 



= x T± (i,j,n) - (K o Q(i,j,n-l)/D 2 )Q T ^(i,j 3 n + l) , (F-2) 



where Q is as defined by equation (54) and D as defined by 

 equation (61) is the arithmetic average of the four values of D 

 about the point i,j at which the flow is evaluated. All terms are 

 spatially centered at i,j and all but the bottom friction terms 

 are exactly centered at time level n . The latter involve flow com- 

 ponents at the new level n+1 and at the old level n-1 within the 

 flow magnitude term Q . This form is known to lead to a stable 

 algorithm (Reid and Bodine, 1968). The Coriolis terms are exactly 

 centered at n as well as i,j . Equations (F-l) and (F-2) repre- 

 sent two equations in the two new Q components Qg*(i,j,n+1) and 

 Qf*(i, j ,n+l) in terms of quantities at previous times and are 

 readily solved for each component individually. The resulting 

 explicit relations for these components are given by equation (55) 

 where G^ , G2 , G3 are defined in equations (58) to (60). 



The centered finite difference analog of the continuity equa- 

 tion (42) leads directly to the explicit relation for H at the new 

 time level given by equation (57) . 



155 



