street^ Chan and Fvomm 



Given Ti(0,y,t) = Ti(t) and Eqs . (13) and (14) and using Eqs. (4-6) 

 at Wall 1 , one can uniquely deternaine u, v and r\ there for the 

 case of a prescribed incoming wave. The conditions at the remaining 

 walls are, of course, not changed. 



The appropriate initial conditions for Eqs. (4-6) are the 

 initial values of the dependent variables, viz. , 



Ti(x,y,0) = 'nj(x,y); u(x,y,0) = Uj(x,y); v(x,y,0) = Vj(x,y) (15) 



3.2. T he Difference Representation and Computation Scheme 



For numerical computation the region 0^x^L|, O^y^Lg 

 is covered by a grid of discrete mesh points with a spacing Ax = Ay = 

 A and calculations are carried forth with time steps At. To allow 

 for proper representation of the boundary conditions the grid indices 

 (i,j) run over the intervals (1,M) and (1,N) respectively and 

 points (i,j), (M,j), (i,l) and (i,N) lie outside the tank walls , e.g., 

 x= is equivalent to (2,j), etc. Consequently, L| = (M-2)A and 

 L2= (N-2)A (see Fig. i). 



In the finite difference representation of the differential 

 equations and auxiliary conditions, central space-differences are 

 always used; both forward and time- splitting schemes are used for 

 time-differences. The differential equations of motion, eqs. (4-6), 

 lead to a highly nonlinear and coupled set of difference equations. 

 These are solved iteratively, using a predictor- corrector method. 

 If u, V and t| are known at all the grid points at tLe n^*^ time 

 level, the following scheme leads to calculation of u, v and r\ ait 

 the n+i^^ level. 



First, u, V and "H are predicted at the n+1 level by use 

 of the nonlinear, shallow water wave equations (Eqs, (4-6) with their 

 right-hand sides set to zero). With the superscript P indicating 

 the predicted value, the difference equations are, after rearrange- 

 ment for computation, 



+ ^\].\ - Vij-|) + Vij(dij.i + ^ij.i - dij., - Tiij.,)) (18) \ 



P At 



^ = ^ij - 2S 



156 



