134
3. Beyondthe One-WayWaveEquation
3.3.3 Stability 01Split Schemes
When the numerical operators :Ft and :F2 commute, the stability of the split
scheme :Ft:F2 is guaranteed by the stability of the individual operators. In order
to demonstrate this, suppose At and A2 are the amplification matrices associated
with:Ft and :F2, and note that if :Ft and:F2 commute, their amplification matrices
also commute. The amplification matrix for the split scheme is AtA2, and
II(AtA2)nll = IIAtA2AtA2 " ·At A211
= II(A»n(A2)nll
s II(At)nllll(A2nl,
(3.62)
(3.63)
and it is apparent that the split scheme inherits the stability properties of the individual operators.
If :Ft and:F2 don 't commute, equality (3.62) does not hold, and
are less than or equal to unity ; however, as discussed in Section 3.1.1, IIAtli
is not necessary for the stability of :Ft .
is the best bound that can be obtained without specific knowledge of At and A2.
The preceding guarantees the stability of the split scheme when IIAJ 11 and IIA211
I
As an illustration of the preceding, consider the system of ordinary differential
equations
du
- = icu + ibu ,
dt
du .
- = I C U ,
dt
(3.64)
(3.65)
where b and c are real constants. Wave solutions to the preceding problem exist
(
) = ( -:/w )Aewhere A is an arbitrary amplitude and w is one of the two real roots to the dispersion relation
w
2
+ bt» - c
2
= o.
A split scheme in which each step is stable but the composite scheme is unconditionally unstable can be obtained by constructing the following finite-difference
approximation to (3.64) and (3.65). In the first step, integrate the terms involving
c using forward-backward differencing:
/).t
_ us
u_ n
. s
= ICU,
/).t
ofthe form
i wt
,