1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 73

vh(x, t), thereby specifying vhT on the grid wh 7 and reducing the error of

approxin1ation to

(x,t) Ewin.

We say that Lh 7 approximates L with order in > 0 in x and with

order n > 0 in t if in the class of sufficiently sn1ooth functions v( x, t) one

of the estimates

is valid with a positive constant JV! independent of I h I and T both.

Example 2

av a^2 v

Lv = - - -

at ax^2 '

0 < x < 1 , 0 < t < t 0 ,

The difference operator Lh 7 defined at all inner nodes of the grid

con1plements subsequent studies and proves to be useful in 111any aspects.

If v( x, t) possesses two derivatives with respect to t and four derivatives

with respect to x ( v E C[t) that are continuous in the rectangle { 0 < x < 1,

0 < t < t 0 }, then

at each of the inner nodes of the grid w hT. This is also consistent with

the results of Section 1.2. Hence, Lh 7 approximates L to second order

in x and first order in t in either of the norms (30) and (31), where

(

N-1 )1/2.
111/J llh = maxxEwh 11/J I or 11 ·1/J llh = Li=l 1/J[ h , etc. Thus, rn the
case of interest the approxin1ation on the grid follows fro111 the local ap-
proximation.
So far we have studied the error of the difference approxin1ation on the
functions v from certain classes V. In particular, in the preceding examples
the class of sufficiently smooth functions stands for V.
In what follows it is supposed that v = u is a solution of a differential
equation, say
L u = u" = -J.