1549301742-The_Theory_of_Difference_Schemes__Samarskii

60 Basic Concepts of the Theory of Difference Schemes

number of the nodes will increase. However, the way of achieving a higher-

order difference approximation we have described above is of little practical

significance, since the quality of resulting operators becomes worse in the

sense of computational volume, the conditions of existence for the inverse

operator, stability, etc. Later in this chapter we will survey some devices

that can be used to obtain higher orders.

In the sequel an auxiliary lemma is needed.

Lemma The formulae are valid:

( 9 ) V,'!:X _ = v(x+h)-2v(x)+v(x-h) h2 = V "(C) <, ' ~=x+eh,

ifv E C(^2 l[x - h, x + h],

(10)

if v E C(^4 l[x - h, x + h]. Here C(k)[a, b] stands for lhe class of functions

with the kth continuous derivative on the seg1nent a < x < b.

Proof We have occasion to use Taylor's formula with the remainder term

in integral form

( 11)

where

v(x) = v(a) + (x - a) v'(a) + · · · + (x -la)" v(r) + R,.+ 1 (x),

r.

x

(12) Rr+i(x) = r\ j (x - ~r v(r+^1 l(O d~

a

1

j(l-s)"v(r+ll(a+s(x-a)) ds.

0

Applying Lagrange's formula to the last integral reveals the remainder term
Rr+1(x) to be