1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for the heat conduction 467

The coefficient at the member Yi becomes non-negative for

( 17)

h2

T < ( 2 1 - CJ) c '

2

which assures us of the validity of the relations

llFJ lie < llYJ lie+ T ll\0^1 lie

and

j

(18) lluJ+

1

llc < llu111c + r ll\OJllc < llu

0

llc +I: T ll\0

11

llc ·

j'=O

Summarizing, the weighted sche111e (9) is stable in the space C, provided

condition ( 17) holds. For the purely implicit scheme with CJ = 1 estimate

(18) is valid for any value of T.

The accurate account of the accuracy is stipulated by more a detailed

exploration of the residual

'ljJ = A (CJ u + ( 1 - CJ) u) + \0 - ut

on the solution u of the original problem (1)-(3). Upon substituting the

expans10ns

1l + U T T - 2

1l = 2 + 2 Ut = fi + 2 ll + 0( T ) ,

1l + U T _ T -; 2

U =

2

• 2 Ut = 1l - 2 U + 0( T ) ,


• 2
  ut = u + O(r )

with the members

. OU
U=-
Oi

it is plain to calculate the residual

=(Au+.f-u)

+(Au - Lu)+ (\0 - .fl+ (CJ - 0.5) r A 1t + O(r^2 ).