1549301742-The_Theory_of_Difference_Schemes__Samarskii

Stability with respect to coefficients 231

To evaluate a solution of problem (4)-(7) on the basis of (6)-(8), we

rely on the estin1ate obtained in Section 3 for ( 4 )-(7), making it possible

to establish

11 z 11 c < 3_ { ( l, I (a - a) Y;; I l + ( l, I 77 I J}.

C1

We shall need the estimates for y and Yx fron1 Section 6 such as

1

//y//c < -(1, /So/),

C1

2

11 Ya; 11 c < - ( 1, I \0 I).

C1

Combination of the inequalities

0 - 0 1 -

( 1, I 771 l < ( 1, / 17 1 J + ( i, Id - d I) 11Y11 c < ( 1, / 17 1 l + - ( 1, I d - d I) ( 1, I \0 I) ,

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0 i-1 0

where 77i = L h(<f>k - \Od, 771 = 0, i = 2, 3, ... , N, and

k=l

( 1, I (a - a) Y;; I] < 3--(1, I \0 I) ( 1, I a - a I l

c1

gives the esti1nate

(9) ll:tJ-yllc<-2{ (1,l^10 71)+-(1,l\Ol)((l,ld-dl)+(l,l2i-all),^2 - }
c1 c1
here Yi is a solution of problen1 (1) and Y; is a solution of problem (2),
provided conditions (3) hold.
Relation (9) can be replaced by a inore rough estin1ate
2 2
(10) 11 :tJ-y lie < - {(1, 110-So 11+-(1, I \0 I) ((1, I cI-d i)+(l, I a-a 1 l)}.
C1 C1
If
0
( 1, I 17 I ) = P( h) , ( 1, I a - a I l = p( h) , ( 1, I cl - d I) = P( h) '
where p(h)---+ 0 ash---+ 0, schemes (1) and (2) are said to be co-equivalent
and for p(h) = O(hm) they are of the mth order of the co-equivalence.
If schemes ( 1) and (2) are co-equivalent and scheme ( 1) is convergent, then
so is scheme (2). This fact follows imn1ediately fron1 the inequality
as h---+0.
The co-equivalence property of homogeneous schemes lies in the main idea
behind a new approach to the further estimation of the order of accuracy
of a schen1e: on account of (9) or (10) its coefficients a, d, \0 should be
cmnpared with coefficients ii, cl, <p of a simple specimen sche1ne, the accuracy
order of which is well-known (see Section 7).