1549301742-The_Theory_of_Difference_Schemes__Samarskii

170 Homogeneous Difference Schemes

When the available functions k(x), q(x) and J(x) happen to be of the class

QC^0 l[o, 1] and their discontinuity points are known, a non-equidistant grid

can be made so that all discontinuity points of the coefficients k, q and

J would be nodal points of such a grid. We denote by wh(K) any such

grid depending on concrete functions k, q and f and arising in subsequent

discussions. It is easily seen from (3) that the simplest expressions for ai,

di and 'Pi on wh(I<) are given by

(oh) ai=ki-1/2> Ci-l - hi q; + 2/i. hi+l qt

z

where Ji± = J(xi ± 0), etc.

ai = -k~+--+-k_-:-_ ,

z-1 z

In the case of continuous coefficients expressions (5) imply that ai = ki_ 112 ,

di = qi and 'Pi = Ji. If discontinuity points coincide with nodal points of

the grid w h, that is, x = xi _ 112 , then the mernbers ai, di and 'Pi can be

found in simplified form:

( 5')

or

2kiki-1

a i--

d i--

'Pi =

2k+ i-1/2 k-:-Z-1/2

k+ k- ,

i-1/2 + i-1/2

hi+l Ji+-112 + hi Ji'.._1/2

2 Iii

'Pi = Ji

2. The error of approxi1nation. We now investigate the error of approx-
   irnation for scheme ( 4) on the non-equidistant grid w h by considering the
   equation for the error z = y - u:

(azx).;; - dz= -1/J(x),
(6)