1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schernes for the heat conduction 465

and Ak(A) is the kth eigenvalue of the operator A. It follows from Green's

formula

that

0

where A y = -Yxx for y E H, with further reference to the relation

(Av,y) = ((y,,)^2 ,1J.

0

The smallest and greatest eigenvalues of the operator A are given by the

formulas

0 4 .,7rh

~ = h2 cos- 2,

0 0

thereby justifying the estimates 15 > c 1 15 and ~ > c 2 ~.

'vVe know from the general stability theory that scheme (10) is stable

in the space HA with respect to the initial data, that is,

under the constraint

In the case where

1 1

Clo= 2 - T ~.

1 l - E

(Jc = 2 - T ~ '

a solution of problem ( 10) satisfies the estin1ate

( 11)

For the explicit scheme ( cr = 0) we might have

( 12 ) Y; j+I -_ ( 1 - h2 T ( a;+ ai+l ))· Y; j + h2 T ( aiYi-l ,j + a;+1Y; j) + Tl.f!;, j

where the coefficient at the member yf is non-negative, provided the con-
dition
2r
1--c h2 2 ->O