456 Stability Theory of Difference Schemesevaluation of the order of approximation. The factor at the member R in
( 108) is T, while in ( 109) is r^2. Therefore, if in the case of two-layer schemes
the condition II Rut II= 0(1) with such a variety of R continues to hold (u
is a solution of the initial differential equation), then the approximation er-
ror changes by a quantity O(r) when R changes. In the case of three-layer
schemes the condition II Rutt II= 0(1) ensures that the regularization leads
to schemes with the approximation error differing by a quantity O(r^2 ). For
this reason three-layer schemes are good enough for the purposes of the
present chapter in designing stable schemes of second-order approximation
ln T.
The main problem here is connected with selecting the regularizator
R. Since regularity conditions became operator inequalities, it seems rea-
sonable to choose as R operators of the most simplest structures which are
energetically equivalent to the operator A. Let, for instance, A and Ao be
energetically equivalent operators with constants / 1 and / 2 , so that
(111) /1 Ao < A < /2 Ao ' 11 > 0, /2 > 0.
Keeping then R = O" Ao we have at our disposal stable schemes: for O" > 0" 0 / 2
(or O" > ~1 2 ) in the case (108) and for O" > 12 /4 in the case (109).
A sin1plest form of R is the operator R = O"E (Ao = E). Stability
conditions are satisfied if O" > O"all A II for scheme (108) and O" > ~II A II for
scheme (109).
Example 1 The explicit three-layer Du Fort-Frankel scheme for the heat
conduction equation from Section 3.7 belongs to the family
(112) yo + (} T^2 Ytt + A Y = 0 ,
t
Indeed,A=A*>O.
Ay = -Ay, Ay = Yxx,
4 2Irh 4
11 A 11 = h 2 cos 2 < h 2 ,that is, the condition O" > tll A 11 is satisfied. This scheme provides a
conditional approxi'mation of O(h^2 ) for T = O(h^2 ).
It is not difficult to write down an explicit stable scheme for the heat
conduction equation with variable coefficients
ou o ( ou)
8t =ox k(x,t) ox '(113) t > 0' O<x<l,
u(O, t) = u(l, t) = 0, u(x, 0) = u 0 (x).