1549301742-The_Theory_of_Difference_Schemes__Samarskii

372 Difference Schemes with Constant Coefficients

Putting these together with ( 4 b) we find that

(26)

which allows us to derive the equation for <1> = j = Y j+i,j r-^1 :

(27) <1> - ()" T^2 A <I> = 'P '

For the purposes of the present section, let us estimate a solution yj of prob-
lem ( 4b) in terms of the right-hand side <p, provided the stability condition
(21) holds. Having stipulated this condition, estimate (20) is certainly true
for a solution of problem (23) and takes for now the form

By the triangle inequality we deduce from (22) that

Equation (27) is needed in obtaining a bound of II Y j'+i,j' 11A_ 1 , in which

of the system {X (kl}:

N-1 N-1

(28) <1>= L <l>kX(k), <p= L'PkX(k).

k=l k=l

Substituting (28) into (27) gives k ='Pk (l + O"T^2 .Ak)-^1. With these rela-
tions in view, we establish for O" > 0

II <1> 11~-l

rneanmg

N-1 <1> 2

2.:::k

k=l ').k

N-I

2.:::

k=l

N-1 2

2.::: ~k = 11'P11~-"

k=I k

II yj'+1,j' llA-1 < T 11 'Pj' llA-1.

Thus, if O" > 0 and condition (21) is fulfilled, the estimate holds for
scheme (4b):