2.4 Space-Differencing
79
schemes, the second-order difference produces a reasonable approximation to the
correct solution. Second-order centered differencing does , however, generate a
noticeable lag in the phase speed ofthe disturbance (as in (2.63)). Moreover, since
the phase lag ofthe 7.51:i.x wave differs from that ofthe 1Ol:i.x wave in the secondorder solution, the relative phase of the two waves changes during the simulation,
and a significant error develops in the amplitude of the two rightmost wave crests.
This example serves to emphasize that although centered differences do not produce amplitude errors in individual Fourier components, they still generate amplitude errors in the total solution. Finally, in contrast to the first- and second-order
schemes, the errors introduced by fourth-order differencing are barely detectable
at this time in the simulation.
The damping associated with the first-order upstream scheme (2.67) can be significantly reduced by using a higher-order one-sided difference. The differentialdifference equation
(2.69)
may be obtained by replacing the spatial derivative in the advection equation with
a third-order difference. Tbc dispersion relation associated with this differentialdifference equation is
cu), = - C [(4 - sinkl:i.x - - sin2kl:i.x
1 ) - -(1 -
i
coskl:i.x) 2] . (2.70)
l:i.x
3
6
3
The real part of CU) , is identical to that of CU4<; , and the phase-speed errors associated
with the third- and fourth -order schemes are therefore identical. As was the case
with first-order one-sided differencing, the sign of the imag inary part of CU) , is
determined by the sign of C such that solutions amplify for c < 0 and damp for
c > O. The damping associated with the third-order sehe me is eonsiderably less
than that of the first-order scheme. Aceording to (2.68) and (2.70),
1
- - = -(1 - coskl:i.x).
3
As might be expected with a higher-order scheme, the well-resolved waves are
damped much more slowly by the third-order approximation. Even the short waves
show substantial improvement.
Some idea of the relative performance of the first-, second-, and third-order
differences is provided in Fig . 2.14, which is identical to Fig . 2.13, except that
the solid curve now represents the third-order solution. As indieated in Fig. 2.14,
the damping produced by the third-order scheme is mueh weaker than that gen -
erated by first-order upstream differeneing. Moreover, the third-order solution to
the spike test is actually better than the second- and fourth-order results (eompare
Figs. 2.13a and 2.14a). In problems with extremely poor resolution, such as the
spike test, the tendency of the third-order seheme to damp short wavelengths can
79
schemes, the second-order difference produces a reasonable approximation to the
correct solution. Second-order centered differencing does , however, generate a
noticeable lag in the phase speed ofthe disturbance (as in (2.63)). Moreover, since
the phase lag ofthe 7.51:i.x wave differs from that ofthe 1Ol:i.x wave in the secondorder solution, the relative phase of the two waves changes during the simulation,
and a significant error develops in the amplitude of the two rightmost wave crests.
This example serves to emphasize that although centered differences do not produce amplitude errors in individual Fourier components, they still generate amplitude errors in the total solution. Finally, in contrast to the first- and second-order
schemes, the errors introduced by fourth-order differencing are barely detectable
at this time in the simulation.
The damping associated with the first-order upstream scheme (2.67) can be significantly reduced by using a higher-order one-sided difference. The differentialdifference equation
(2.69)
may be obtained by replacing the spatial derivative in the advection equation with
a third-order difference. Tbc dispersion relation associated with this differentialdifference equation is
cu), = - C [(4 - sinkl:i.x - - sin2kl:i.x
1 ) - -(1 -
i
coskl:i.x) 2] . (2.70)
l:i.x
3
6
3
The real part of CU) , is identical to that of CU4<; , and the phase-speed errors associated
with the third- and fourth -order schemes are therefore identical. As was the case
with first-order one-sided differencing, the sign of the imag inary part of CU) , is
determined by the sign of C such that solutions amplify for c < 0 and damp for
c > O. The damping associated with the third-order sehe me is eonsiderably less
than that of the first-order scheme. Aceording to (2.68) and (2.70),
1
- - = -(1 - coskl:i.x).
3
As might be expected with a higher-order scheme, the well-resolved waves are
damped much more slowly by the third-order approximation. Even the short waves
show substantial improvement.
Some idea of the relative performance of the first-, second-, and third-order
differences is provided in Fig . 2.14, which is identical to Fig . 2.13, except that
the solid curve now represents the third-order solution. As indieated in Fig. 2.14,
the damping produced by the third-order scheme is mueh weaker than that gen -
erated by first-order upstream differeneing. Moreover, the third-order solution to
the spike test is actually better than the second- and fourth-order results (eompare
Figs. 2.13a and 2.14a). In problems with extremely poor resolution, such as the
spike test, the tendency of the third-order seheme to damp short wavelengths can