334 Difference Schemes with Constant CoefficientswhereXN-i(xi) =sin 7r(N - 1) X; =sin 7r(N - 1) ih
=sin 7r(l - h) i = (-1)'-^1 sm JrXi
so thatThis provides support for the view that the solution is completely dis-
torted. From such reasoning it seems clear that asymptotic stability of a
given scheme is intimately connected with its accuracy. When asymptotic
stability is disturbed, accuracy losses may occur for large values of time.
On the other hand, the forward difference scheme with O" = 1 is asymp-
totically stable for any T and its accuracy becornes worse with increasing
tj, because its order in t is equal to l. In practical implementations the
further retention of a prescribed accuracy is possible to the same value for
which the explicit scheme is applicable. Hence, it is not expedient to use
the forward difference scheme for solving problem (1) on the large tirne
intervals.
- The scheme of second-order accuracy (unconditionally stable in the as-
ymptotic sense). Before taking up the general case, our starting point is
the existing scheme of order 2 for the heat conduction equation possessing
the unconditional asyrnptotic stability and having the form
(8)
Twhere Ay = Yxx> E is the identity operator, yj+i/^2 is the intermediate
value and O" = 2 - /2. For i = 0 and i = N the hon1ogeneous boundary
conditions are specified by
(9)