1549301742-The_Theory_of_Difference_Schemes__Samarskii

506 Homogeneous Difference Schemes for Tiine-Dependent Equations

Theorem Let the functions k(x, t) and f(x, t) have discontinuities of the

first kind on a finite number of the straight lines x = ~s, s = 1, 2, ... , s 0

parallel to tlie axis Ot and in the regions of the special configurations

6.s=(~s<x<~s+ 1 ,0<t<t 0 ), s=0,l, ... ,s 0 , ~ 0 =0, ~so+i=l,

the coefficients k( x, t) and f ( x, t) and the solution u( x, t) are smooth enough

so that both conditions (19) and (15) hold true. Then under condition (22)

scheme (7)-(9) converges uniformly with the rate O(r^2 + h^2 ) on special

sequences of nonequidistant grids w 1 J K) and the solution of problem ( 11)

satisfies the estiinate

(25) where h 0 = n1ax h;.

l<i<N

To prove this assertion, it suffices to bring together a priori estimates

(21)-(22) with relations (15) and (19).

Re1nark The theorem is still valid upon replacing (7) by the schen1e

(26)

2 0

( E - (J T A ) Ytt = A y + 'P ,

0

where the constant operator Ay = Yxx is adopted as a regularizer. In that

0

case A = -A, R = -(J A and D = E + r^2 R. The sufficient stability

condition (21) is ensured if we agree to consider

(27)

In evaluating the error of approximation all the tricks and turns remain
unchanged except for fonnula (14) for T), in which the member (JT^2 aitft5J
should be replaced by (JT^2 1lftx, where a constant (J is specified by (27).