1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General formulations 125

by the difference operators A 1 y = Yxx + b Yx for b > 0 or A 2 y = Yxo: + b Yx

for b < 0 with y E rt h.

~ 0

Let A y be an op era~or from Q h into ~ h coinciding with A y for y E Q h.

The operators A 1 = -A 1 and A 2 = -A 2 , acting from Q h into Q h, are

positive definite for any h. Indeed,

(19)

( A1 y, Y) = ( -vtx, Y) - b ( Y,,.' Y) = ( 1 + ~ h b) 11Yxl12'

( A2 u, u ) = ( 1 - ~ h b ) 11 u x JI 2 = ( 1 + ~ h I b I ) 11 v" J I 2.

We conclude frmn here that

for O'.= 1, 2.

Be re-ordering A 1 = A+ b A+ and A 2 = A+ I b I A - , where A = -A,

Ay = Yxx and II A± II < 2 / h, II A II < 4 / h^2 , we are led due to the triangle

inequality for the norms to

11 A 1 11 < 11 A 11 + b 11 A+ 11 < : 2 ( 1 + ~ h b ) , b >^0 J

(20)

II A2 II < II A II+ b II A- II < h

4

2 (^1 + ~ h I b I), b < 0.

We note in passing that if we approximate the operator L u = u" + bu' by

the expression A y = Y-xx + b Y-x for b > 0, then the member 1 - 21 - h b arises

in place of 1 + ~ hb in (19) and thereby the operator -A will be positive

definite only for-h < 2 / b.

If u'( x) is approximated by the central difference derivative ux for

any sign b, we have the operator A 3 y = -A 3 y, where A 3 y = Yxx + b Yx·

This operator A 3 y = -yxx - h b R 3 y is of second-approximation order and

satisfies the relations

and

Although the complete theory could be recast in the general case, we
confine ourselves to simplest exarnples. In subsequent chapters the differ-
ence operators approximating elliptic operators (in particular, the Laplace