1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 57

form, that is, to compose the set of nodes adjacent to x at which the values

of the grid function v( x) are aimed at approximating the operator L.

In this section we consider several exam pies of difference approxima-

tions for elementary differential operators.

Example 1 Lv = dv. Let us fix some point x on the Ox-axis by capturing

dx

the neighboring points x-h and x+h, where h > 0, and try to approximate

Lv. Also, it will be sensible to introduce the following expressions:

(1) Ltv v(x+h~-v(x) vx,

(2)

_ _ v( x) - v( x - h)

L1i v = h v,,,.

Expressions (1) and (2) are called the right difference derivative

and the left difference derivative and are denoted by vx and V;;:, respec-

tively. The difference expressions Lt and L~ are defined at two points. In

other words, these are based on the two-point patterns x, x +hand x - h,

x. Moreover, any linear combination of expressions (1)-(2) such as

(3)

where (} is a real number, can be adopted as the difference approximation

of the derivative dv/dx. In particular, for (} = 0.5 we get the central

two-sided difference derivative

1 v( x + h) - v( x - h)

(4) Vo x = -2. (vx. + Vx-) ' = 2h

It turns out that there is an uncountable set of difference expressions ap-

proximating Lv = v' and this is something one might expect. The following

question is of significant importance: what is the error of one or another dif-

ference approximation and how does the difference 1/J( x) = L1i v( x) - Lv( x)

behave at a point x ash----+ 0? The quantity 1/J(x) = L1i v(x) - Lv(x) refers

to the error of the difference approximation to Lv at a point x. We

next develop v( x) in the series by Taylor's formula

h2

v(x ± h) = v(x) ± h v'(x) +

2

v"(x) + O(h^3 ),

assuming v( x) to be a sufficiently smooth function in some neighborhood
( x - h 0 , x + h 0 ) of the point x and h < h 0 , where the number h 0 is kept