1549301742-The_Theory_of_Difference_Schemes__Samarskii

424 Stability Theory of Difference Schemes

After that, taking into account that

Yi-1 = -h Yx ,i +Yi , Yi+1 = h Yx ,i +Yi , h Yx ,i - h Yx ,i = h

2

Yxx ,i ,

Yi-l = Yi-l + TYt,i-1 = Yi-l -hYx,i + TYt,i - hrY.i:t,i,

substituting these expressions into (72) and omitting the subscript i, we

come to

(73) w TYt = h^2 Y.i:x - CY h TYxt.

Dividing (73) by h^2 generates

(7 4)

CYT

Yt + h Yxt = Yxx ·

0

2) Let H h be the space of grid functions D h (see Examples 1 and 2

in Chapter 2, Section 4.1) defined on the grid wh ={xi= ih, 0 < i < N}

under the inner product structure (y, v) = 2={:~^1 Yi vi h. In conformity with

the results obtained in Section 4.1 of Chapter 2, the operators Ay = -Yxi·

and R1y = ~Yi: involved in the sche1ne are positive definite: (R 1 y, y) =

~(Ay, y). The operator A is self-adjoint, 11 A II < 4/h^2.

3) The operators A and R 1 are constant. Because of this, it will be

convenient to write sche1ne (74) in the form

(75) (E +CY T Ri) Yt + Ay = 0 J

so that B = E + CYrR1.
The condition B > ~rA is satisfied for CY> 1-2/(rllAll). Indeed,
for any x EH

yielding

((B - 0.5rA)x,x) = ((E+ CYrR 1 - 0.5rA)x,x)

= ((E+0.5r(CY-l)A)x,x),

B-05rA>E+0.5r(et-l)A> (ll~ll +0.5r(CY-l))A>O.

4. Since llAll < 4/h^2 , scheme (71) is stable in the space HA (in the
   grid norm of the space vVd-) for

(76) CY>

h2

1--.

2r