1549301742-The_Theory_of_Difference_Schemes__Samarskii

552 Economical Difference Schemes for Multidirnensional Problerns

where B = (E + 0.5 T Ai) (E - 0.5 T A2), Aa = -Aa, A= Ai+ A2.

Because of the rectangular form of the initial domain, the operators

Ai and A 2 are self-adjoint, positive and coinmu tative. It is straightforward

to verify the relations AiA2 = A2Ai and AiA2y = A2AiY = Yi·i:uix 2 ,, 2 at all

the inner nodes of the grid. In view of this, the strict inequality AiA 2 > 0

is simple to follow. From the form (24) it seems clear that

(25) B > E + 0.5 T,

thereby clarifying that scheme (24) is stable in the space HA. Indeed,

(

T r

2

B - 0.5 r A= E + - A+ - A ) T

2 4 1 A2 - - 2 A

From condition (25) it follows that for scheme (24) Theorem 7 from Chapter

6, Section 2 is still valid with E = 1, due to which a solution of proble1n

(22) satisfies the inequality

(26) llY( t + T JI I A < I ly(O)ll A + ;, (i:;, T 11 ~(t') 11' )"'

Also, the a priori estimate holds true:

(27) II y(t + r) II < II y(O) II+ ;, (i:;, T ll~(t')llA ') </2

To inake sure of it, we apply the operator A-^1 > 0 to both parts of equation
(24). The outcome of this

• 
  
  
  
  • B Yt + Ay = (/;, A-E - '
    (28) 2
    B - = A -i + -T E + -T A -i A1 A2.
    2 4
  
  
  
  

Since the operators Ai, A 2 and A-i > 0 are commutative and self-adjoint,
the relations A-iAiA 2 > 0 and B > A-^1 +0.5 rE occur. Applying Theo-
ren1 10 from Chapter 6, Section 2 yields estimate (27).