86 Basic Concepts of the Theory of Difference SchemesExample 4. The three-layer difference scherne for the heat con-
duction equation. A special attention is being paid to the first boundary-
value problem(52)au 82 u
Ft= 8x2 + .f(x, t)' U < x < 1 ,^0 < t < t^0 ,u(O, t) = u 1 (t), u(l, t) = u 2 (t), u(x, U) = u 0 (x).
The usual practice in numerical analysis of the heat conduction equa-
tion (52) is connected with three-layer sche1nes. The values yj-^1 (x), yj(x)
and yj+^1 (x) of a grid function on the three time layers tj_ 1 , tj and tj+i
are aimed at constructing such schemes.
The three-layer symmetric scheme on the equidistant grid w In with
steps h and r, being the most familiar one, comes first:(53)Y 0 j = uj 1 J YN j = uj 2'
where Ay = Yxxi u is a real parameter and <pj = .f(x;,tj).
Since the central difference derivative in t approximates to the sec-
ond order in T ~~ lt=t 1 and t,.u = ~:~ + O(h^2 ), scheme (53) approximates
equation (52) with accuracy O(h^2 + r^2 ). From what has been said above
it is clear that problem (53) is overdetermined as it were. Applications
of the three-layer scheme concerned necessitate imposing one more initial
condition, for instance, by doing this on the first layer. But under such a
condition the approximation 0( r^2 + h^2 ) should remain unchanged. There
seem to be at least two ways of determining y( x, T). One way of proceeding
is to approve at the first step the two-layer scheme of accuracy 0( r^2 + h^2 )
in specifying y( x, T):
Y1 _Yo 1
___ = _A (y1 +yo)+ 'Po.
T 2
Under the second approach the value of y(x, r) arranges itself as a sum
y(x, r) = u 0 (x) + rμ(x) andμ is so chosen as to obtain the error y(x, r) -
u(x, r), not exceeding 0( r^2 + h^2 ). Substitution of the value ~~ lt=O' arising
from the differential equation
au[ -
8=Lu 0 +.f(x,O),
t t=O