1549301742-The_Theory_of_Difference_Schemes__Samarskii

504 Homogeneous Difference Schemes for Time-Dependent Equations

A comparison of (7) and (20) provides enough reason to conclude that these

operators A and D are identical:

A = -A , D = E - (J r^2 A = E + (J r^2 A.

The operator A is self-adjoint., positive definite and satisfies the estimate

(see Chapter 2, Section 4)

The general stability theory outlined in Chapter 6 asserts that scheme (20)

is stable under the condition

l+c- 2 l+c-

D >

4

T A or (Dy, y) >

4

r^2 (Ay, y),

where E > 0 is an arbitrary number independent of h. We will pursue the

further stability analysis of this with

D- 1 + 4 E T^2 A=E+ ( (J-^1 +4 E) T2 A

(

> w+^1 ( (J-^1 + E) 2)

4 T A>o,

which is certainly true for

h2. n11n

4 T^2 C. 2

If you wish to explore this more deeply, you might find it helpful to
refer to Chapter 6, Section :3 of the monograph "The Theory of Difference
Schemes", in which the following estin1ates were derived for problen1 (20):

(21) llzj+
1
llA(t;) < M 187 (llzt(O)llnc 7 J + t^7 ll1/iklln-^1 (tk)),

(22) llzj+
1
llA(tjl < lvf 187 (llzt(O)lln( 7 )