184 Homogeneous Difference SchemesThe natural replacement of the central difference derivative u' ( x) by
the first derivative ua x leads to a scheme of second-order approximation.
Such a scheme is monotone only for sufficiently small grid steps. More-
over, the elimination n1ethod can be applied only for sufficiently small h
under the restriction h Ir( x) I < 2 k( x). If u^1 is approxin1ated by one-sided
difference derivatives (the right one ux for r > 0 and the left one ux for
r < 0), we obtain a monotone scheme for which the maximum principle is
certainly true for any step h, but it is of first-order approximation. This is
unacceptable for us.
It is worth mentioning here that the sign of r( x) has had a significant
impact on construction of monotone schemes. One way of providing a
second-order approximation and taking care of this sign is connected with a
monotone scheme with one-sided first difference derivatives for the equation
with perturbed coefficients
(20) Lu= -f, Lu = x ( k u^1 )^1 + r u' - q u ,
where x = 1/(l+R) and R = 0.5 h Ir l/k is the Reynolds difference number.
By obvious rearranging r(x) as a sumr ~ (r - Ir I) :::; 0,
the expression ru' is approximated by the forn1ula(ru'); = (~ (ku')) i ~ bt Cl;+ 1 ux,i + bj Cl; ux, i,
where bt = F [r±(x;+sh)], :;::± = r± /k, Fis a pattern functional being used
for calculations of the coefficients d and <p. We may accept, for instance,
b+ = r+ / k and b- = r-/ k. As a result we get the homogeneous scheme
Ay = x(ayx)x + b+ a(+l)Yx + b-a Yx - dy = -<p,
(21) y 0 = u 1 ,. YN = u 2 , a(+l) = a(x + h), a 2 c 1 > 0,
v _ 1 R _ L:__L!:
re- l+R' - 2k.\'Ve are going to show that scheme (21) is monotone by observing that
(22) Ai Yi-1 - Ci Yi + B; Yi+1 = l.f!i ,
where
B i -- v a;+1 ( X; +I I b+) i , Ci = A; + Bi + d;.