1549301742-The_Theory_of_Difference_Schemes__Samarskii

162 Homogeneous Difference Schemes

helps motivate what is clone with 1/J; = (qi 1l; - d, Hi) + ( <p; - Ji) + 0( h^2 ) =

O(h^2 ).

We now turn to the case of discontinuity coefficients k(x), q(x) and

J(x) and assume without loss of generality that k, q and f have disconti-

nuities of the first kind only at a single point x = ( E (0, 1), so that

( = Xn +eh, 0 < e = 8(h) < 1 J 0 < n < N.

The solution 1l = H( x) to equation (1) for x = ( satisfies the continuity

conditions for the function H(x) and the flow k(x)H'(x):

[1l] = 1l(( + 0) - 1l(( - 0) = 0, [k1l^1 ] = 0 for X = (.

Under the assurnptions k(x), q(x), f(x) E QC^2 ) we thus have u(x) E QC^3 l.

Before going further, it. is straightforward to verify for 17; that

17; = (aux); - (kH^1 );_ 112 = O(h^2 ) for all if:. n + 1.

In dealing with T/n+i = an+i 1l:r,n+i - (ku')n+1/2 we rnake use of the expan-

sions in a neighborhood of the point x = (

(1 8)^2

1ln+i = u(() + (1-8) h 111 (( + 0) + ~ h^2 H^11 (( + 0) + O(h^3 ),

1

{ (kll')x=~+o + (0.5 - 8) h (k1l

1

)~=~+o + O(h^2 )

(kit )n+l/2 =

(ku')x=~-o + (0.5 - e) h (ku')~=~-o + O(h^2 )

fore< 0.5,

fore> 0.5,
by means of which an alternative forrn of writing 77n+i can be re-ordered
for later use:

T/n+l = an.+1 ( (1 - 8) ll~ight + e u;eft) - Wa + 0( h) >

where w 0 = (ku')right = (k1t')1eft, v 1 ert = v(( - 0) and vnght = v(( + 0).

At the sarne time, the continuity condition [kv.'] = 0 assures us of the

validity of the relations

(1 - e )Uright / + e Uleft / = (e -k-+ 1-8) k. Wa '

left nght

T/n+1 = [an+1 (ke. + ~ -e)-1] w 0 +O(h).

lett nght