1549301742-The_Theory_of_Difference_Schemes__Samarskii

58 Basic Concepts of the Theory of Difference Schemes

fixed. Substituting the preceding series into (1), (2), and (4) yields

v= v(x + h) - v(x) =vx+-vx+^1 ( ) h^11 ( ) O(h^2 )

x h 2 ,

(5) V:r; = v(x) - ~(x - h) = v1(x) - ~ v11(x) + O(h2),

Vo = v(x + h) - v(x - h) = V^1 ( X ) + O(h^2 )

x 2h ,

thereby justifying that

·1/J = vx - v^1 (:r) = O(h),

1jJ = vx - v^1 (x) = O(h),

1/J=v~ -v^1 (x)=O(h^2 ).

Let V be a class of sufficiently smooth functions v E V defined in a

neighborhood Patt(x, ho) of a point x containing for h < h 0 the pattern

Patt(x, h) of a difference operator Lh. We say that Lh approximates the

differential operator L with order m > 0 at a point x if

1/J(x) = Lh v(x) - Lv(x) = O(hm).

Thus, the right and left difference derivatives generate approximations of
order 1 to Lv = v^1 , while the central difference derivative approximates to
the second order the same.

d^2 v

Example 2 Lv = v^11 = dx 2. In order to construct a difference approxi-

n1ation of the second derivative, it is necessary to rely on the three-point
pattern ( x - h, ;c, x + h). In that case we have

(6)

v( x + h) - 2 v( x) + v( x - h)

Lh v = h 2

Observe that the right difference derivative at a point x is identical
with the left difference derivative at the point x + h, that is, the relation
vx(x) = va;(x + h) occurs, permitting us to rewrite (6) as