358 Difference Schemes with Constant Coefficients- Explicit schemes of a higher-order approximation. Of major importance
is the explicit scheme of accuracy O(h^2 + r) having the form
(7)or Yt + ay'i: = 0, where y'i:,i = (Y;+ 1 -y;_ 1 )/(2h). This scheme is con-
structed on the four-point pattern (Fig. l 7c) consisting of the four nodes
(x;, tj+ 1 ), (xi, tj), (xi_ 1 , tj) and (xi+1> tj)· Obviously, scheme (7) is un-
stable for every fixed I = a T h-^1 and arbitrary sign of the coefficient a.
Indeed, upon substituting harmonic (4) into equation (7) we get the equa-
tion for q:i = V-1, q =^1 - i I sin <p.
Whence it follows that I q I 2 = 1 + 12 sin^2 <p > 1, so that I y 11 = I q I j --+ oo
as j --+ oo. Adopting the arguments similar to those used for the scheme of
Section 1.10 and replacing Yf by the half-sum! (Y1+i + yi_i), we obtain
the stable scheme on the same pattern:(8)for which
Yk j +1 -- 2 1 (^1 +I ) Yk-1 j + 2 1 (^1 - I ). Yk+1' j
II yj+I lie < II yj lie < ... < II Y^0 lie
for any I I I < 1 and arbitrary sign of the coefficient a. More specifically,
for a < 0 we thus have 1 + / = 1 - I/ I > 0 and 1 - / = 1 + I/ I > 0.
In the estimation of the residual for scherne (8) it is possible to rewrite
it as
0.5 h^2
Yt - T Yxx + a Y'i: = 0
in light of the trivial relations
2 1 ( Yk+1+Yk-1 ) --2 1 ( Yk+1+Yk-1-2 ) Yk +Yk-Yk+2 -^1 h2 Yxx,k·