3.5 Linear Equations with Variable Coefficients
153
The analogous result for the averaging scheme is most easily obtained by noting
that
({cJ8x4Jjr = [82x(Cj4Jj) + Cj 82x4Jj - 4Jj 8 2x Cj]
= iCk
l ak2 (sin[(k) +
+ sin
- sink)
)e i (k l + k 2 )jtu .
(3.100)
In the limit of good numerical resolution, k) S» ,
0, and each of the
above expressions is equivalent, which is to be expected, since (3.99) and (3.100)
are both second -order approximations to (3.98). As (k) +
approaches n ,
the numerical formulae may become inaccurate, but the most serious problems
develop when 7f < I(k) +
longer wavelength .
27f and the product term is aliased into a
Suppose that a wave of wave number k2 in the 4J field is interacting with some
disturbance in the velocity field to force d4J/dt at an aliased wave number k.
According to (3.91), there is only one resolvable wave number in the velocity
field that could alias into k through interaction with k2 during the approximation
ofthe product co1/J/ox. The rate at which this aliasing occurs can be computed as
folIows. Without loss of generality, assurne that the unresolvable wave number is
positive (i.e., k) + k2 > 7f /
in which case k is negative and
(3.101)
Suppose that at a given instant both interacting waves have unit amplitude, i.e.,
ICk11 = lak 2 1 = 1. Let C k2
-+ k = d lakl/dt denote the rate at which interactions
between the wave numbers k) and k2 force the growth at the aliased wave number.
Using (3.101) to eliminate k) from (3.100), one obtains a growth rate for the
averaging scheme of
A
C
- =
I
+
- sin(k -
The growth rate for the nonaveraging scheme,
N
Isink2ßxl
C
- =
,
k2-+ k
can be obtained directly from (3.99).
Sx
.
A contour plot of
as a function of ka and kappears in Fig. 3.11a.
Contours of the wave number k) involved in these interactions (computed from
(3.10 I)) also appear plotted as a function of (k2, k) as the dashed diagonallines in
Fig. 3.11. Since k) 7f /
(because every wave must be resolved on the numerical mesh), no aliasing can occur for the (k2, k) combinations above the diagonal
in Fig. 3.11a, and this region of the plot is left blank. Equivalent data, showing
contours of
for the averaging scheme, appear in Fig. 3.11b.
As indicated in Fig. 3.11, the nonaveraging scheme allows every wave number
on the resolvable mesh (0
k2
n /
to combine with some disturbance
153
The analogous result for the averaging scheme is most easily obtained by noting
that
({cJ8x4Jjr = [82x(Cj4Jj) + Cj 82x4Jj - 4Jj 8 2x Cj]
= iCk
l ak2 (sin[(k) +
+ sin
- sink)
)e i (k l + k 2 )jtu .
(3.100)
In the limit of good numerical resolution, k) S» ,
0, and each of the
above expressions is equivalent, which is to be expected, since (3.99) and (3.100)
are both second -order approximations to (3.98). As (k) +
approaches n ,
the numerical formulae may become inaccurate, but the most serious problems
develop when 7f < I(k) +
longer wavelength .
27f and the product term is aliased into a
Suppose that a wave of wave number k2 in the 4J field is interacting with some
disturbance in the velocity field to force d4J/dt at an aliased wave number k.
According to (3.91), there is only one resolvable wave number in the velocity
field that could alias into k through interaction with k2 during the approximation
ofthe product co1/J/ox. The rate at which this aliasing occurs can be computed as
folIows. Without loss of generality, assurne that the unresolvable wave number is
positive (i.e., k) + k2 > 7f /
in which case k is negative and
(3.101)
Suppose that at a given instant both interacting waves have unit amplitude, i.e.,
ICk11 = lak 2 1 = 1. Let C k2
-+ k = d lakl/dt denote the rate at which interactions
between the wave numbers k) and k2 force the growth at the aliased wave number.
Using (3.101) to eliminate k) from (3.100), one obtains a growth rate for the
averaging scheme of
A
C
- =
I
+
- sin(k -
The growth rate for the nonaveraging scheme,
N
Isink2ßxl
C
- =
,
k2-+ k
can be obtained directly from (3.99).
Sx
.
A contour plot of
as a function of ka and kappears in Fig. 3.11a.
Contours of the wave number k) involved in these interactions (computed from
(3.10 I)) also appear plotted as a function of (k2, k) as the dashed diagonallines in
Fig. 3.11. Since k) 7f /
(because every wave must be resolved on the numerical mesh), no aliasing can occur for the (k2, k) combinations above the diagonal
in Fig. 3.11a, and this region of the plot is left blank. Equivalent data, showing
contours of
for the averaging scheme, appear in Fig. 3.11b.
As indicated in Fig. 3.11, the nonaveraging scheme allows every wave number
on the resolvable mesh (0
k2
n /
to combine with some disturbance