1549301742-The_Theory_of_Difference_Schemes__Samarskii

88 Basic Concepts of the Theory of Difference Schemes

which can be rewritten as

1

Y;=SY;-1, s=--- 1 +ah ' i = l, 2, ... , Yo = U 0 ,

yielding Yi = si Yo.

We regard a point x to be fixed and take a sequence of steps h so

that x would always belong the set of grid nodes: x = i 0 h. The attached

number i 0 may be made arbitrarily large once we will refine the grid in any

convenient way, that is, letting h --+ 0. The value of y at this point becomes

y(x) = Yio = sio Yo·

Since Is I < 1 for a > 0 and any h, we thus have

for any h.
The last inequality implies that the solution of the difference problem
(2) continuously depends on the input data. In such cases we say that a
difference schen1e is stable with respect to the input data.

Example 2. An unstable scheme. For problem (1) we rely on the

scheme

(3)

Yo = Uo' i = 1, 2, ... '

where u > 1 is a numerical parameter. Observe that this scheme is a three-
point one, since the difference equation is of order 2, so there is some reason
to assign the value y 1 in addition to the value Yo. The approximation order
of scheme ( 3) is no less than 1 regardless of the choice of the parameter u.
With u 0 = ( 1 - ah )u 0 , a proper evaluation of the deviation gives ii, - u( h) =
O(h^2 ). We look for particular solutions to the difference equation (3) in
the form Yi = si. Substituting Yi = si into (3) gives the quadratic equation
related to an unknown s:

(4) ( u - 1) s^2 - (2u - 1 +ah) s + u = 0,

which possesses two distinct roots

2 u - 1 +ah± Jl + 2 (2u - 1) ah+ ct^2 h^2

s 1 - ----------------~

'^2 - 2(u-l)