90
2. BasicFinite-Difference Methods
We now consider finite-difference approximations to the complete partial differential equation and analyze the total error that arises from the combined effects of
both temporal and spatial differencing.
In some instances, the fundamental behavior of a scheme can be deduced from
the characteristics of its constituent spatial and temporal differences. For example,
suppose that the advection equation
-+c-=o
at/J
at/J
at
ax
(2.86)
is approximated using forward time-differencing in combination with centered
spatial differencing. The result should be amplifying because forward time -differencing is amplifying and centered spatial differencing is neutral. Their combined
effect will therefore produce amplification. On the other hand, it might be possible
to combine forward time-differencing with one-sided space-differencing because
the one-sided spatial difference is damping-provided that it is computed using
"upstream" data. If this damping dominates the amplification generated by the
forward time difference, it will stabilize the scheme. Further analysis would be
required to determine the actual stability condit ion and the phase-speed error.
As another example, consider the use of leapfrog time-differencing and centered spatial differencing to approximate the advection equation. Since both differences are neutral, it seems likely that such a scheme would be conditionally stable. Once again, further analysis is required to determine the exact stability condition and the phase-speed error. In the absence of such analysis, the sign of the
phase-speed error is in doubt, since the leapfrog scheme is accelerating, whereas
centered spatial differencing is decelerating. Finally, suppose that leapfrog differencing is combined with one-sided spatial differences. The result should be
unstable because the leapfrog solution consists of two modes (the physical and
computational modes) each propagating in the opposite direction. Ifthe one-sided
difference is "upstream" with respect to one mode, it will be "downstream" with
respect to the second mode, thereby amplifying the second mode.
Although as just noted , the forward -time and centered-space scheme
2L\x
ö.t
A,n+ I
'l'j
A,n
- 'l'j
A,n
A,n
=
0
'l'j+1 - r t-:
(2.87)
generate convergent approximations to the correct solution in the limit L\x
0,
L\t
0, because it is a consistent approximation to the advection equation. Rewill produce a nonphysical amplification of the approximate solution to the advection problem, one might wonder whether this amplification is sufficiently weak
that the scheme nevertheless satisfies the more general Von Neumann stability
condition
(2.88)
where y is a constant independent of k, M, and Sx, If so, then (2.87) will still
call that as discussed in Section 2.3.2, forward differencing produces amplifying
2. BasicFinite-Difference Methods
We now consider finite-difference approximations to the complete partial differential equation and analyze the total error that arises from the combined effects of
both temporal and spatial differencing.
In some instances, the fundamental behavior of a scheme can be deduced from
the characteristics of its constituent spatial and temporal differences. For example,
suppose that the advection equation
-+c-=o
at/J
at/J
at
ax
(2.86)
is approximated using forward time-differencing in combination with centered
spatial differencing. The result should be amplifying because forward time -differencing is amplifying and centered spatial differencing is neutral. Their combined
effect will therefore produce amplification. On the other hand, it might be possible
to combine forward time-differencing with one-sided space-differencing because
the one-sided spatial difference is damping-provided that it is computed using
"upstream" data. If this damping dominates the amplification generated by the
forward time difference, it will stabilize the scheme. Further analysis would be
required to determine the actual stability condit ion and the phase-speed error.
As another example, consider the use of leapfrog time-differencing and centered spatial differencing to approximate the advection equation. Since both differences are neutral, it seems likely that such a scheme would be conditionally stable. Once again, further analysis is required to determine the exact stability condition and the phase-speed error. In the absence of such analysis, the sign of the
phase-speed error is in doubt, since the leapfrog scheme is accelerating, whereas
centered spatial differencing is decelerating. Finally, suppose that leapfrog differencing is combined with one-sided spatial differences. The result should be
unstable because the leapfrog solution consists of two modes (the physical and
computational modes) each propagating in the opposite direction. Ifthe one-sided
difference is "upstream" with respect to one mode, it will be "downstream" with
respect to the second mode, thereby amplifying the second mode.
Although as just noted , the forward -time and centered-space scheme
2L\x
ö.t
A,n+ I
'l'j
A,n
- 'l'j
A,n
A,n
=
0
'l'j+1 - r t-:
(2.87)
generate convergent approximations to the correct solution in the limit L\x
0,
L\t
0, because it is a consistent approximation to the advection equation. Rewill produce a nonphysical amplification of the approximate solution to the advection problem, one might wonder whether this amplification is sufficiently weak
that the scheme nevertheless satisfies the more general Von Neumann stability
condition
(2.88)
where y is a constant independent of k, M, and Sx, If so, then (2.87) will still
call that as discussed in Section 2.3.2, forward differencing produces amplifying