Stability of a difference scheme^89It is well-known that the general solution to equation (3) is of the form
(5) Yi=As~+Bs~.
Putting i = 0 and i = 1 and using the assigned values Yo = 110 and y 1 = ll 0 ,
it is easy to calculate the constants A and B:B=
It is clear that s 1 s 2 > 1, since u > u - 1 > 0. We claim that s 2 < 1
for any value of ah. Indeed, for u > 1
2 (u - 1) - (2u - 1 +ah - Jl + 2 (2u - 1) ah+ a2h2)= J(l +ah )^2 + 4 ( u - 1) ah - ( l +ah) > 0.
Let us stress also that s 1 > 1 for any value of ah on account of the bound
s 1 s 2 > 1 in view.
From the well-established representation Yi = As~ + B s~ it is easily
seen that Yi --+ = as i--+ = if A op 0. But we can always select y 1 = u 0
so that the condition A = 0 holds true for the choice u 0 = u 0 s 2. The
rounding errors arise inevitably in developing the solution s~, thus causing
the instability of the indicated type. In this scheme the solution increases
along with increasing X; = ih if h is kept fixed. Successive grid refinement,
that is, successive refinement of h, leads to the growth of errors at a fixed
point x = i 0 h, since i 0 = x / h increases along with decreasing h. A small
change of input data results in an enormous change in the solution of the
problem at any fixed point x as h --+ 0.
We quote below the results of computations for problem (3) with
Yo = 1 and y 1 = s 2 , where s 2 is the smallest root to the quadratic equation
( 4). Once supplemented with those initial conditions, the exact solution of
problem (3) takes the form Yi =Es~ (A= 0). Because of rounding errors,
the first summand emerged in formula (5). This member increases along
with increasing i, thus causing abnormal termination in computational pro-
cedures.
Modern computers allow the implementation of model problems. We
have carried out the calculations for several variants:
(J" = 1.1, ah= 0.01, S1 = 11.11, S2 = 0.99;
(J" = 1.1, ah=O.l, S1 = 12.09, s 2 = 0.91;
(J" --^2 J ah= 0.01, Si = 2.02, 82 = 0.99;
(J" --^2 ' crh = 0.1, 81 = 2.17, 82 = 0.92.