1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 59

With the aid of the expansion of the function v(x) in Taylor's series it is

not difficult to show that the order of approximation is equal to 2, meaning

vxx - v^11 (x) = O(h^2 ) by virtue of the expansion

(8)

h2

v-xx = v^11 + - 12 v(^4 ) + O(h^4 ).

Example 3 Lv = v(^4 ). We now deal with the five-point pattern consisting

of the points (x - 2h, x - h, x, x + h, x + 2h) and accept Lh v = V:cxxx· By

applying formula (6) to V:rx we derive the expression for vxxxxi which will

be needed in the sequel:

= : 4 [v(x+2h)-4v(.T+h)+6v(x)-4v(x-h)+v(x-2h)].

It is straightforward to verify that Lh provides an approximation of order

2 to L, so that Vxxxx - v(^4 ) = h 6

2

v(^6 ) + O(h^4 ). Indeed, this can be done

using the expansion in Taylor's series

(^7) (J"s ks hs
v(x + O"kh) = v(x) + L 1
s=l S.
dsv(x) O(h8)
dxs + ' (]" = ±1 ,
fork= 1, 2 and exploiting the obvious fact that the sum v(x+kh)+v(x-kh)
contains only even powers.
The expansion of the approximation error 1jJ = Lh v - Lv in powers
of h is aimed at achieving the order of approximation as high as possible.
Indeed, we might have
II 13!' (4) ( 4) h
2
Vxx V - ( 4)
12
V + 0 h -
12
V:lix:lix + 0 h ,
whence it follows that Lv = v^11 is approximated to fourth order by the
operator
on the pattern (x - 2h, x - h, x, x + h, x + 2h).
In principle one can continue further the process of raising the order
of approximation further and achieve any order in the class of sufficiently
smooth functions v E V. During this process the pattern, that is, the total