1549301742-The_Theory_of_Difference_Schemes__Samarskii

670 Methods for Solving Grid Equations

upon substituting

From such reasoning it seems clear that II sk II = Tk/2 - 1 for tk < 0 and

II sk II= 1 - Tk/1 for tk > 0. If tk < 0 and k > ko, where ko is the minimal

number for which tko < 0 and p 0 (1+2lt1.: 0 I) > 1, then

This provides enough reason to conclude that

n

II II Sj II >II ska 11n-ko > 1

j =ko

and rounding errors in specifying Yko will grow with increasing k from k 0

to n.

Let~=^11 ~1,sothatp 0 =l-2~+0(~^2 )andtk =cos^2 A;i-l7f<0.

~ • n

All this enables us to deduce that

Under the assumptions lt1.:, I> 0.5 and~< 0.01, we might have

11sk,11=3(1-60 + 0(~^2 ) > 3. o.9 = 2.1,

n

II II sj II> 2.1n-kl.

j =k1

As a n1atter of fact, formulas (30)-(31) express the stability of scheme
(14) with parameters (29) with respect to the initial data. Those ideas
supported by the preceding calculations provide proper guidelines for the
further stability analysis of con1putational procedures with respect to the
right-hand side as well as with respect to the initial data in passing from
y 0 to Yk for any k = 1, 2, ... , n. From the general stability theory outlined
in Chapter 6, Section 1 we recall that stability with respect to the right-
hand side is a corollary to the uniform stability with respect to the initial
data, meaning stability in the process of moving fron1 any Yj to any Yk with
k > j > 0.