590 Economical Difference Schemes for Multidimensional ProblemsObserve that the primary scheme Ytt + r^2 RYrt
under the restriction(} = E = const > 0.
By replacing
p
E + T^2 R = E + T^2 L Rex
ex=l
by the factorized operatorp
D =II (E+r^2 Rex)
ex= 1we obtain the econon1ical factorized sche1nepA y + 1.p is stable
II(E+r^2 Rex)Yrt=Ay+i.p for .rEwh, tEw 7 ,
ex=ly = μ for x E /h, t E w 7 ,
y(x, 0) = u 0 (x), Yt(x, 0) = ii 0 (x) for x E wh,
where ii 0 (x) = ii 0 + 0.5 r(Lu 0 + f(x, 0)).
It is plain to show that the scheme concerned is absolutely stable and
it generates an approximation of order 2: l/J = O(r^2 + lhl^2 ), v = O(r^2 ).
Whence the convergence with the rate O(r^2 + lhl^2 ) immediately follows.
The search for yj +^1 amounts to successive solution of three-point equa-
tions of the fonn ( E + r^2 Rc~)w = F ex by the elimination method for every
component of the vector w with the index account from CY to Cl'+ 1. One
possible way of covering this is connected with the perfonnance of the fol-
lowing algorithm:
p
( E + r^2 Ri)wl 1 l = F , F = II ( E + r^2 Ro:) Yr + r (A y + 1.p) ,
a=l( E + T^2 R " ) w(o:J = W(cx-1), '-' A• =^2 '· · · ' ' p yj +^1 = yj + T w (p)'