1549301742-The_Theory_of_Difference_Schemes__Samarskii

472 Homogeneous Difference Schemes for Tirne-Dependent Equations

arising from the representation

0

( 36) 1/J = 7),r + 1/i* , that is, ·!/J = 7), 0 ,

gives rise to a perfect model for a more simpler estimation:

( :37)

for i < n + 1,

for i > n + l.

The order of accuracy of sche111e (7) can be most readily evaluated

with the aid of the representation for the error z = y - 11 as

z -- v + z* ,

where v and z* are solutions of the related problems:

0

vt = A (er v + ( 1 - er) v) + ijJ , v 0 =vN=0, v(x,0)=0,

z; = A(cri +(l -cr)z) +1/,, z Q = Z~, 1Y = (^0) ) z(x,0)=0.
In the accurate account of v and z we apply the results of the general
stability theory (Theorems 9 and 11 from Section2, Chapter 6):
j
+2:::rllA-^1 ~fll for cr>J 0 ,
J'=I
j
(39) 11z]+I11 < I: T 111/J]' II for er> cr 0 , er> 0.
j'=O
0 0
Since A-^1 ijJ ( is a solution to the equation A( 1/J T/x, we have
occasion to use the relation
( 40)
the right-hand side of which can be modified on account of (37) into
N
(1, IT/I]= I: h l17il = h^2 l1/Jnl + h Ii/Jn+ 1/Jn+1I (1-Xn+l) ·
i=n+l