666 Methods for Solving Grid EquationsIn what follows problem (37) will be treated as a model one in the
further con1parison of various methods in a step-by-step fashion in line with
established priorities and answering real needs. We concentrate primarily
on the total number of the iterations required in the simple iteration method
(34)-(34') and the method with optimal set ofChebyshev's parameters (14),
(29).
Recall that it is fairly common to write the iteration number k over
the sought function y within the frameworks of iterative methods available
for difference equations. The same procedure works in the simple iteration
scheme (SIS) which has been designed for pro bl em (37):
k+l k k
(39) Y =y+r 0 (Ay+f),
where
To=
fl+ f2 4
Upon substituting the assigned value into (39) we derive the formula
k k k k
( 4 0) k+l Yi1-l,i2 + Yi1+l,i2 + Yi1,i2-l + Y;^1 h+l h(^2) .f
y i1 i2 = 4 + 4 i1 i2 ,
by means of which the (k + l)th iteration is c0111pleted.
For the explicit scheme (14) with Chebyshev's parameters (SCP) the
calculations are performed by the formula
k+l k Tk+l
Yi1i2 = Yi1i2 + 7
where
k
Since the volumes of computations in determining y in these methods
differ slightly, the main criterion in such matters is the total number of the
iterations. By formula (38) we find for h ~ 1 that
fl .,7rh 7r2h2
~ = - = tg--~ --
f2 2 4
For example, the reasonable accuracy c = 2 e-^10 ~ 10-^4 is attained for
n 0 (c) ~^3 fi^2 in the case of SCP and for n 0 (c) ~ ;; 2 in the case of SIS,
making it possible to fill in the following table regardless of the initial
approximation, that is, for any element y 0 of the space H.