1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 67

In a similar manner one can wt·ite down on the pattern 5.a the operator

(24)

On the other hand, a two-parameter family of difference operators

is based on the nine-point pattern 5.d, whence (23) follows for 0" 1 = 0" 2 = 0

and (24) follows for 0" 2 = 0 and 0" 1 = 1. We have occasion to use the

asymptotic formulae

82 v(x, t) 2

V11 = 8t2 +O(r ), vxx - = a2v(x, 8x2 t) + 0(12) i ,

thereby justifying that operator (23) provides an approximation of O(h^2 +

r^2 ). Operator (25) has the same order of approximation for 0" 1 = 0" 2 = O",

where O" is an arbitrary number.

We note in passing that a key role of the parameters 0" 1 and 0" 2 just

as the parameter O" in the previous example is connected not only with the

approximation order, but also with stability of the appropriate difference

scheme. This important property will appear in subsequent discussions in

Chapter 5, Section L

Example 6. L v = v^11 • An irregular pattern (a non-equidistant grid).

Grantee! two numbers h_ > 0 and h+ > 0, other ideas are connected with

the three-point pattern ( x - h_, x, x + h+). When h_ f- h+, any such

pattern is said to be irregular, since the grid including this pattern turns

out to be non-equidistant. Being concerned with new members

v(x)-v(x~h_)

vx = h_ ,

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

Vx = h

+

we refer to the operator L1i with the values

Ii= h_ + h+

2

(26) L1i v - ~ Ii [v(x + h+) h+ - v(x) v(x) - v(x h_ - h-)]

For h_ = h+ = n. the preceding is identical with expression (7) (see Example
2). Plain calculations of the local approximation error at a point x show
that