310 Difference Schemes with Constant CoefficientsUnder the constraint (} > () 0 it is not difficult to verify that m:;x I qk I < 1
and 111111 < llYll orII yj+i II< II Yj II < .. · <II Y^0 II= II Uo II·
If so, the solution of problem (16a) satisfies the estimate
(24) j = 1, 2, .. , I
which rneans that schen1e (16) is stable in the grid L 2 (wh)-norm with respect
to the initial data for(}> () 0.
A difference scheme is said to be conditionally stable when it is
stable only if T and h are related in some way, otherwise it bec0111es un-
conditionally stable. A scheme, stable for arbitrary values of T and h,
is said to be absolutely stable. One can encounter schemes stable for
sufficiently small h and T such that h < h 0 and T < T 0. These schemes are
not absolutely stable, but may be unconditionally stable.
Throughout the entire chapter, a special attention is being paid to
several types of schemes.
- The explicit scheme((}= 0). Condition (23) gives 0 > ~ - h^2 (4r)-^1 ,
that is,
(25)T 1
< -
h^2 2The explicit scheme is stable only under condition (25) relating the grid
steps hand T (a conditionally stable sche111e).
- For(} > ~the implicit scheme is stable for any hand r, since(} > ~ > () 0.
Thus, the forward difference sche1ne ( (} = 1) and the sy111metric scheme
((}=~)are stable for any hand T, what means, by definition, their absolute
stability. - The scheme of a higher-order approximation ((} = (},, (}* = ~ -
h^2 (12r)-^1 ) is absolutely stable. Indeed, for any hand T
h2
> 0.
6T- Implicit schemes with 0 < (} < ~for (}independent of I= r/h^2 are
conditionally stable if one imposes the constraint / < (2 - 4 (} )-^1. - Scheme (16) with (} = ~ + h^2 CY r-^1 providing an approximation of
O(h^2 +t^2 ) is stable for any hand T if CY>-~.