1549301742-The_Theory_of_Difference_Schemes__Samarskii

The transfer equation 361

Observe that scheme (11) belongs to this family, corresponds to the case

O" = 1 and has the residual 1/; = ut + O" ilx + ( l - O") ux. By inserting here

the asymptotic expansions

ux = u I - 2 1 I 1 'tl II + O(h2) '

V = V - ~ TV+ 0( r^2 ),

where v = v/t 1 +T/ 2 and v = ur, we arrive at

1/J = ~ T (2 O" - 1) U

1

• ~ h U^11 + 0( h^2 + r^2 )

which serves to motivate that the scheme with weights is of order 2: 1/; =

O(h^2 + r^2 ) if O" = ~ - ~ h T-l = 0" 0. In all other cases (O" f::. 0" 0 ) it is of order

1, since 1/; = O(r+h).

In order to demonstrate that scheme (12) is stable with respect to

initial data for

1 h

O">---=O" -2 2T O>

we proceed as usual. This arnounts to forming the grid wh = { X; = i h,

i = 0, 1, ... , N, hN = l} on the segment 0 < x < l and introducing the

inner product and associated norm by

N

( y, v] := L Yi vi h, //y// := J(y, y].

i=l

Other ideas are connected with the well-established expressions

1 1

v= -(v+v)+-(v-v)

2 2

and

v= ~(v+v)-~(v-v)

2 2

and forthcoming substitution v = Yx· Upon receipt of the available infor-
mation we reduce the scheme concerned to

Yt + ( O" - ~) T Yxt + ~ (i}r + Yr) = 0 , y(O, t) = 0.