1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes^409

provided that the forthcoming substitutions are carried out before going

further:

1) Xn = B^112 yn for B = B* > 0, where C = C1 = B-^1 /^2 AB-^1!^2 ;

2) xn = A^112 yn for A= A*> 0, where C = C2 = A^112 B-^1 A^112.

Definitions 1) and 2) together imply that

llxnll=llYnlls forC=C1,

II Xn II = II Yn llA for C = C2,

x n = B1f2y n,

x n = Alf2y n·

The condition of p-stability of the implicit scheme (la) in the space HD for

the choices D = B and D = A is equivalent to the condition of p-stability

of the explicit scheme (32) in the space H: II xn II< Pnll X 0 II, n = 1, 2, ....

Lemma 2 Let scheme (32) be given with a constant operator C. The

condition of p-stability of this scheme is equivalenl to the boundedness of

the norm of the transition operator

II s II = II E - Tc II < p.

Indeed, we have x 1 = Sx 0 and II X 1 II < II S II · II X 0 II for n = 1. By

comparing this inequality with (31) for n = 1 we finish the proof of Lemma

2.

Lemma 3 If A= A > 0 and B = B > 0, then the inequalities

(33)

are equivalent for the operators C = B-^1 /^2 AB-^112 and C = A^112 B-^1 A^112.

Proof Let C = B-^1 /^2 AB-^112 and I be an arbitrarily taken number. The
difference

(Cx, x) -1 (x, x)'= (B-^1 /^2 AB-^1 l^2 x, x) -1 (x, x) = (Ay, y) -1 (By, y)

with regard toy= B-^1!^2 x shows that the signs of the operators C -1E
and A - 1B coincide. No property of positiveness of the operator A is
required here.
Let now C = A^1 /^2 B-^1 A^1!^2. First, we are going to establish the
equivalence of the inequalities C > / E ( C < / E) and E > / c-^1 ( E <
,c-^1 ). Upon substituting y for C^1 l^2 x we obtain

( Cx, x) - / ( x, x) = ( C^112 x, C^112 x) - / ( x, x) = (y, y) - I ( c-^1 y, y),