SECT. 5] TIOES 787 



3. To start the computation not only boundary values are required but also 

 initial values. If friction is taken into account, the final solutions — that means 

 f, u, V after a sufficiently long time — become independent of these initial values. 



The process to set up a system of difference equations is the following. The 

 sea under consideration, G, is covered by a grid with mesh-size I. All derivatives 

 in space direction x,y are replaced by central differences according to (10). 

 Derivatives in time direction are substituted by forward differences with time 

 interval t, 



m f{t + r)-f{t) 



et ^ T 



Introducing these finite differences into the equations (3) for a grid point 

 with co-ordinates x, y at the time t, it follows that 



u{t + T, X, y) = ( 1 — T • i?<^) )u{t, X, y) -fTv{t, x, y) 



-Yi[^\t,x + l,y)-^'{t,x-l,y)l 



v{t + T,x,y) = [l-T ■B<y))v{t,x,y)+fTu{t,x,y) 



-YiW{t,x,y + l)-^'{t,x,y-l)l 



hr 



^{t + T,x,y) = - — {u{t, x + l,y)- u{t, x-l,y) + v{t, x,y + l)- v{t, x,y- 1)] + 



i{t, X, y) 

 for all x, y of grid points. 



The following abbreviations will now be introduced : a vector Z{t), its 

 components consisting of all ^, u, v, a matrix A{t) arising from the system and 

 a further vector C, of which the components are determined by the boundary 

 values and the external forces. It follows that : 



Z{t + r) = Z{t)-T-A{t)-Z{t) + T-C{t). (11) 



This equation points out the characteristics mentioned above. Starting with 

 the known data at time t, the distribution of Z — that means of |, u, v in all 

 grid points — at time ^-|-t is available without solving a system of equations. 

 After n time steps the result is, assuming A and C to be independent of time t, 



Z{t + nT) = Z{t){I-T-A)n + [I + {I-TA)+ . . . +(/-t^)«-i]t-C. 



/ is the unit matrix. 



This solution is only valid in the case of convergence ; that means the dif- 

 ference between Z and the exact solution of the system (3) must be small. 

 Kreiss (1958, lO.lBa. 1959, 1959a) has shown that each Cauchy problem for the 

 system of linear differential equations properly posed can be solved approximately 

 by a stable system of difference equations. This is not only valid for one or two 

 dimensions but also for more than two. It is, therefore, possible to tackle tidal 



