7.3 Fractional-Step Methods
367
2
e
r
0
u
2
e
FIGURE 7.4. Spectral radius of the amplification matrix for the partially split method
contoured as a function of cand ufor (a) M = I (b) M = 3. Unstable regions are enclosed
in the wedged-shaped areas. Contour intervals are 1.0 (heavy line), 1.2, 1.4, . .. Une AB
indicates the possible combinations of cand uthat can be realized when U [c = 1/10 and
M is specified as I or 3.
the possible combinations of uand cwilllie along a straight line of slope
u Ußt
U
- = - = M - .
c cS»
e
Suppose that U / e = 1/ IO. Then if the partial splitting method is used with M =
I, the only possible combinations of uand care those Iying along line AB in
Fig. 7Aa. The maximum stable value of ßT is determined by the intersection
of the line AB and the left boundary of the leftmost region of instability. Thus,
for U/ e = 1/ lO and M = I, the stability requirement is that cbe less than
approximately 1.25.
As demonstrated in Fig. 7Ab, which shows contours of the spectral radius of
SÄ 6 , the restrietion on the maximum stable time step becomes more severe as
M increases to 3. The regions of instability are narrower and the strength of the
instability in each unstable region is reduced, but additional regions of instability appear, and the distance from the origin to the nearest region of instability
decreases. When M = 3 and U / e = 1/ I0, the maximum stable value of cis
roughly 0048. Further reductions in the maximum stable value for coccur as M
is increased, and as a consequence, the gain in computational efficiency that one
might expect to achieve by increasing the number of small time steps per large
time step is eliminated by a compensating decrease in the maximum stable value
for ß T.
The partial splitting method has, nevertheless, been used extensively in many
practical applications. The method has proved useful because in most applications it is very easy to remove these instabilities by using a filter. As noted by
Tatsumi (1983) and Skamarock and Klemp (1992), the instability is efficiently
367
2
e
r
0
u
2
e
FIGURE 7.4. Spectral radius of the amplification matrix for the partially split method
contoured as a function of cand ufor (a) M = I (b) M = 3. Unstable regions are enclosed
in the wedged-shaped areas. Contour intervals are 1.0 (heavy line), 1.2, 1.4, . .. Une AB
indicates the possible combinations of cand uthat can be realized when U [c = 1/10 and
M is specified as I or 3.
the possible combinations of uand cwilllie along a straight line of slope
u Ußt
U
- = - = M - .
c cS»
e
Suppose that U / e = 1/ IO. Then if the partial splitting method is used with M =
I, the only possible combinations of uand care those Iying along line AB in
Fig. 7Aa. The maximum stable value of ßT is determined by the intersection
of the line AB and the left boundary of the leftmost region of instability. Thus,
for U/ e = 1/ lO and M = I, the stability requirement is that cbe less than
approximately 1.25.
As demonstrated in Fig. 7Ab, which shows contours of the spectral radius of
SÄ 6 , the restrietion on the maximum stable time step becomes more severe as
M increases to 3. The regions of instability are narrower and the strength of the
instability in each unstable region is reduced, but additional regions of instability appear, and the distance from the origin to the nearest region of instability
decreases. When M = 3 and U / e = 1/ I0, the maximum stable value of cis
roughly 0048. Further reductions in the maximum stable value for coccur as M
is increased, and as a consequence, the gain in computational efficiency that one
might expect to achieve by increasing the number of small time steps per large
time step is eliminated by a compensating decrease in the maximum stable value
for ß T.
The partial splitting method has, nevertheless, been used extensively in many
practical applications. The method has proved useful because in most applications it is very easy to remove these instabilities by using a filter. As noted by
Tatsumi (1983) and Skamarock and Klemp (1992), the instability is efficiently