This, together with the equation dy/dt = 3Q/dx, is solved in a cyclic 
scheme. One possible method is the centered Crank-Nicolson type implicit- 
explicit scheme discussed below. Suppose y is given for all x at a given 
time, t, and that from t the wave climate is specified by the (constant) 
triple (a5H), 1). Let 
Ny ie mi EQ Ot Eh 
( xd) = dag ee (8¥/ax)° 
At 
eres 
L(t,x) = an approximation to L(t,x) 
Integrating the Q equation gives 
At 92Q 97Q 
OG <2 Neyo) = OlGeno) mal L(t,x) aoe IGE av Att, x) aa (46) 
it ANE 
where 
OQ OCs hd = 206) = WGe= iss 
oS eee ee ee ee (47) 
ox Ax 
Integrating the y gives 
At [3Q IQ 
QB S ING559) = 37(E5) Sea] Oe (48) 
2 \\ee aX 
t + At 
where 
30 _ Q(x + Ax) = Qe = 4x) (49) 
OX 2Ax 
These equations are solved numerically, subject to the appropriate boundary 
conditions, in steps using the cyclic algorithm: 
(a) Wee 16Ge 2 Wes) S Ges Wee, 
(b) Calculate Q(t + At,x) Vx subject to the appropriate boundary 
conditions. 
(c) Calculate y(t + At,x) Vx; calculate L(t + At,x) and set this 
equal to L(t + At,x); then calculate new Q. 
(d) If new Q compares with old Q, stop; if not go to step (c). 
Tests with this scheme have shown that it converges to its limit after one 
application of step (c). 
This method can easily be modified to solve the equation where both dif- 
fraction and variations in lake level are allowed; i.e., m is the slope 
(refer to eqs. 37 to 40). 
OB 1 0 
= ED cos a, sin ap mars (50) 
(28) 
