122
3. Beyond the One-Way WaveEquation
where that ray intersects the dashed curve labeled "3." As indicated in Fig. 3.4a,
the nonaveraging scheme does not resolve the propagation of 2.t.s waves parallel
to either the x- or the y-axis, but 2.t.s waves can move at greatly reduced speed
along the diagonalline x = y (0 = n / 4). The phase-speed error diminishes as the
wavelength increases, with the maximum error in 6.t.s waves being no larger than
20% . These results can be compared with the relative phase speed curves for the
averaging scheme plotted in Fig. 3.4b. The averaging scheme generates substantial errors in the phase speed of waves moving diagonally along the line x = y;
the 2.t.s wave does not propagate at all, and even the 6.t.s wave is significantly
retarded. These reduced phase speeds allow the averaging scheme to remain stable for large time steps, but as is apparent in Fig. 3.4, the enhanced stability is
obtained at the cost of increased phase-speed errors in the poor and moderately
resolved waves.
Forward-in-TIme Schemes
Assuming that U ::: 0 and V ::: 0, one generalization of the upstream method to
the two-dimensional advection equation (3.20) is
(3.31)
The stability of this scheme may be investigated using the standard Von Neumann
method. Let
Then
»J
wm.n -
_ Ajei(kmßxHnßY) .
(3.32)
A = I - /L(I -
- v(l - e- i S),
(3.33)
where as before, /L = U.t.t/.t.x , v = V.t.t/.t.y, g = k Sx , and
Necessary and sufficient conditions for stability are
= l.t.y.
Os v,
and
(3.34)
The necessity of the preceding may be established by considering the three cases
g = 0, = 0, and g = for each of which the dependence of the amplification
factor on the wave number reduces to an expression of the same form as in the
one-dimensional case (2.25). The sufficiency of (3.34) follows from
lAI S 1I - /L - v] +
+ Ive-isl
= 1I - /L - v] + I/LI + [v],
which implies that lAI s I whenever /L and v satisfy (3.34).
Suppose that Sx = .t.y = .t.s and that C is abound on the magnitude of
the two-dimensional velocity vector; then provided that the spatial differences are
evaluated in the upstream direction, the stability condition is c.t.t /.t.s s l/..ti.
As was the case with the leapfrog approximation (3.21), the maximum stable
time step is approximately 30% less than that in the analogous one-dimensional
3. Beyond the One-Way WaveEquation
where that ray intersects the dashed curve labeled "3." As indicated in Fig. 3.4a,
the nonaveraging scheme does not resolve the propagation of 2.t.s waves parallel
to either the x- or the y-axis, but 2.t.s waves can move at greatly reduced speed
along the diagonalline x = y (0 = n / 4). The phase-speed error diminishes as the
wavelength increases, with the maximum error in 6.t.s waves being no larger than
20% . These results can be compared with the relative phase speed curves for the
averaging scheme plotted in Fig. 3.4b. The averaging scheme generates substantial errors in the phase speed of waves moving diagonally along the line x = y;
the 2.t.s wave does not propagate at all, and even the 6.t.s wave is significantly
retarded. These reduced phase speeds allow the averaging scheme to remain stable for large time steps, but as is apparent in Fig. 3.4, the enhanced stability is
obtained at the cost of increased phase-speed errors in the poor and moderately
resolved waves.
Forward-in-TIme Schemes
Assuming that U ::: 0 and V ::: 0, one generalization of the upstream method to
the two-dimensional advection equation (3.20) is
(3.31)
The stability of this scheme may be investigated using the standard Von Neumann
method. Let
Then
»J
wm.n -
_ Ajei(kmßxHnßY) .
(3.32)
A = I - /L(I -
- v(l - e- i S),
(3.33)
where as before, /L = U.t.t/.t.x , v = V.t.t/.t.y, g = k Sx , and
Necessary and sufficient conditions for stability are
= l.t.y.
Os v,
and
(3.34)
The necessity of the preceding may be established by considering the three cases
g = 0, = 0, and g = for each of which the dependence of the amplification
factor on the wave number reduces to an expression of the same form as in the
one-dimensional case (2.25). The sufficiency of (3.34) follows from
lAI S 1I - /L - v] +
+ Ive-isl
= 1I - /L - v] + I/LI + [v],
which implies that lAI s I whenever /L and v satisfy (3.34).
Suppose that Sx = .t.y = .t.s and that C is abound on the magnitude of
the two-dimensional velocity vector; then provided that the spatial differences are
evaluated in the upstream direction, the stability condition is c.t.t /.t.s s l/..ti.
As was the case with the leapfrog approximation (3.21), the maximum stable
time step is approximately 30% less than that in the analogous one-dimensional