788 HANSEN [chap. 23 



problems as well as special problems of dynamical oceanography with this 

 method. It is important to note that this theorem given by Kreiss is only valid 

 for linear systems. Nevertheless, it can be anticipated that, as long as the non- 

 linear terms are not dominant, the behaviour of the solution is similar to that 

 in the linear case. Special caution is necessary in cases of non-linear equations. 

 A very simple example may explain the difference method and give also 

 some idea of its accuracy. If there is only one grid point, only one ordinary 

 differential equation exists, with the unknown function F : 



^^ + aF{t) = Ce^yi. 

 ct 



li t = nr the solution is given by : 



C \ Ce^y^'' 

 F{nT) = e-<'»'\Fo--. 4- 



ly + aj ly + a 

 with F{0) = Fq. The difference equation is 



F'[{n+l)r] = {l-ar)F'{nT) + rCeiy^\ 

 With these values expressed by the initial value Fo, it follows that 



F'[{n +l)r] = {l-ar)r^+-^-Fo + Creiy^n_^ ^^TTlV 



The accuracy of F' is given by 



1 T 



F{nT)-F'{nT) = Ce^y^^ 



n 1 p-auT f ci-rpivT ^ 



{\-ary 



+ 





ly + a 



l-[(l-aT)/e'>] 



This difference converges to zero when r becomes small, the real component of 

 a = ai + i-a2 as a friction term is positive and T<2ail{ai'^ + a2^). The result is 

 that a suitable choice of t makes it possible to approximate the exact solution 

 of the differential equation by means of difference methods. In the more general 

 case of (11), the condition of approximation is that the matrix must be a normal 

 one, and the absolute value of eigen values must be smaller than one. Kreiss 

 has shown that it is possible to fulfil this condition in each properly posed 

 problem taking the mean of values in neighbouring grid points. In some cases 

 it is sufficient to take t< l|2^/{gh). 



8. Numerical Solutions of Initial-Boundary- Value Problems of Tides in One 



and Two Dimensions 



It appeared appropriate at first to apply this difference method to one- 

 dimensional problems. There have been some calculations made for tidal 

 estuaries ; in these regions the equations are non-linear. At the mouth of the 



