7. Physically Insignificant Fast Waves
366
-iie,
UHa
- - 2
xx
I
h(
-ü»,
I
-ga x
1+ c
2a
xx
2
where
0
0
0
).
-ü»,
0
0
0
0
I
and the tilde denotes multiplieation of the parameter by Ö.r (e.g., c = cÖ.r) .
At the beginning of the first small time step in a eomplete big-step-small-step
integration eyc1e
At the end of the (2M)th small step
Thus, if S is a matrix interehanging the first pair and seeond pair of elements in r ,
(
S=
I 0 0 0
o I 0 0
'
the eomplete big-step-small-step integration eyc1e is given by
0
o 0
0 0 I 0) I
Sinee the individual operators eommute, the cornpletely split approximation to
this problem is stable whenever both of the individual fractional steps are stable.
One might hope that the stability of the partially spilt mcthod could also be guaranteed whenever the large- and small-step subproblems are stable. Unfortunately,
there are many eombinations of ö.t and Ö.r for which the partially split method
is unstable even though the subproblerns obtained by setting either U or c to zero
are both stable (Tatsumi 1983; Skamaroek and Klemp 1992). Suppose that the
partially split seheme is applied to an individual Fourier mode with horizontal
wave number k. Then the amplifieation matrix for an individual small time step
is given by a matrix in which the partial -derivative operators in A are replaced by
ik; let this matrix be denoted by A.
Consider the case M = I, for whieh the amplifieation matrix is SA 2 . The
magnitude of the maximum eigenvalue , or speetral radius p, of SA 2 is plotted in
Fig. 7Aa as a funetion of c = ck txt and u = Uk St, The domain over whieh
p is contoured, 0 SeS 2 and 0 S u Si, is that for which the individual
small- and large-step problems are stable . When M = I, P exceeds unity, and the
partially split scheme is unstable throughout two regions of the c-u plane whose
boundaries interseet at (c, u) = (../2, 0). If U « c, only a limited subset of the
c-u plane shown in Fig. 7Aa is actually relevant to the solution of the shallowwater problem . Onee the number of small time steps per large time step is fixed,
366
-iie,
UHa
- - 2
xx
I
h(
-ü»,
I
-ga x
1+ c
2a
xx
2
where
0
0
0
).
-ü»,
0
0
0
0
I
and the tilde denotes multiplieation of the parameter by Ö.r (e.g., c = cÖ.r) .
At the beginning of the first small time step in a eomplete big-step-small-step
integration eyc1e
At the end of the (2M)th small step
Thus, if S is a matrix interehanging the first pair and seeond pair of elements in r ,
(
S=
I 0 0 0
o I 0 0
'
the eomplete big-step-small-step integration eyc1e is given by
0
o 0
0 0 I 0) I
Sinee the individual operators eommute, the cornpletely split approximation to
this problem is stable whenever both of the individual fractional steps are stable.
One might hope that the stability of the partially spilt mcthod could also be guaranteed whenever the large- and small-step subproblems are stable. Unfortunately,
there are many eombinations of ö.t and Ö.r for which the partially split method
is unstable even though the subproblerns obtained by setting either U or c to zero
are both stable (Tatsumi 1983; Skamaroek and Klemp 1992). Suppose that the
partially split seheme is applied to an individual Fourier mode with horizontal
wave number k. Then the amplifieation matrix for an individual small time step
is given by a matrix in which the partial -derivative operators in A are replaced by
ik; let this matrix be denoted by A.
Consider the case M = I, for whieh the amplifieation matrix is SA 2 . The
magnitude of the maximum eigenvalue , or speetral radius p, of SA 2 is plotted in
Fig. 7Aa as a funetion of c = ck txt and u = Uk St, The domain over whieh
p is contoured, 0 SeS 2 and 0 S u Si, is that for which the individual
small- and large-step problems are stable . When M = I, P exceeds unity, and the
partially split scheme is unstable throughout two regions of the c-u plane whose
boundaries interseet at (c, u) = (../2, 0). If U « c, only a limited subset of the
c-u plane shown in Fig. 7Aa is actually relevant to the solution of the shallowwater problem . Onee the number of small time steps per large time step is fixed,