5.5 Flux-Limiter Methods
269
(a)
, \
\ i
\i
.
\
\ \
x
2
3
I
/
o
306
' I
;'
/
,, ----l
x
FIGURE 5.13. Comparison of MC ftux-limited (long-dashed line) and FCT (solid line)
approximate solutions with the exact solution (thin dashed-dotted line) for the two test
cases shown in Fig. 5.12.
most completely free from phase-speed error, and as a consequence the overall
character of the f1ux-limited solution is superior to that obtained with the LaxWendroff method (i.e., with no limiter), which exhibits a substantial phase lag
and very modest damping .
Figure 5.13 shows a comparison of the MC flux-limited scheme with the Zalesak f1ux-corrected transport algorithm discussed in Section 5.4.3. The two test
problems are identical to those just considered in Fig. 5.12. Both methods are
implemented using the same methods to evaluate the monotone and high-order
f1uxes (specifically upstream differencing and the Lax-Wendroff method) . The
solutions obtained with the MC f1ux-limited method are clearly superior to those
obtained using the FCT scheme. The tendency of the FCT scheme to deform the
sine wave into a sawtooth can, however, be eliminated using a second iterative
pass of the FCT algorithm as discussed at the end of Section 5.4.2 (Schär and
Smolarkiewicz 1996, Fig. 4).
Since f1ux-correctedtransport and f1ux-limiter methods both revert to first-order
schemes in the vicinity of minima and maxima, they do not give fully secondorder approximations in problems like the sine-wave-advection test shown in
Figs. 5.12b and 5.l3b. The effective order of accuracy of these schemes can be
empirically determined for the sine-wave-advection test by performing aseries
of simulations in which both tu and I1t are repeatedly halved (so that all simulations are performed with the same Courant number of 0.5). Fitting a function
of the form a(l1x)P to the error as I1x decreases from 1/40 to 1/320 yields the
approximate values for plisted in Table 5.1. As a check on the quality of this
calculation, the empirically determined orders of accuracy for the upstream and
Lax-Wendroff schemes are also listed in Table 5.1. The result for the upstream
scheme is slightly in error, and could be improved by continuing the computa-
269
(a)
, \
\ i
\i
.
\
\ \
x
2
3
I
/
o
306
' I
;'
/
,, ----l
x
FIGURE 5.13. Comparison of MC ftux-limited (long-dashed line) and FCT (solid line)
approximate solutions with the exact solution (thin dashed-dotted line) for the two test
cases shown in Fig. 5.12.
most completely free from phase-speed error, and as a consequence the overall
character of the f1ux-limited solution is superior to that obtained with the LaxWendroff method (i.e., with no limiter), which exhibits a substantial phase lag
and very modest damping .
Figure 5.13 shows a comparison of the MC flux-limited scheme with the Zalesak f1ux-corrected transport algorithm discussed in Section 5.4.3. The two test
problems are identical to those just considered in Fig. 5.12. Both methods are
implemented using the same methods to evaluate the monotone and high-order
f1uxes (specifically upstream differencing and the Lax-Wendroff method) . The
solutions obtained with the MC f1ux-limited method are clearly superior to those
obtained using the FCT scheme. The tendency of the FCT scheme to deform the
sine wave into a sawtooth can, however, be eliminated using a second iterative
pass of the FCT algorithm as discussed at the end of Section 5.4.2 (Schär and
Smolarkiewicz 1996, Fig. 4).
Since f1ux-correctedtransport and f1ux-limiter methods both revert to first-order
schemes in the vicinity of minima and maxima, they do not give fully secondorder approximations in problems like the sine-wave-advection test shown in
Figs. 5.12b and 5.l3b. The effective order of accuracy of these schemes can be
empirically determined for the sine-wave-advection test by performing aseries
of simulations in which both tu and I1t are repeatedly halved (so that all simulations are performed with the same Courant number of 0.5). Fitting a function
of the form a(l1x)P to the error as I1x decreases from 1/40 to 1/320 yields the
approximate values for plisted in Table 5.1. As a check on the quality of this
calculation, the empirically determined orders of accuracy for the upstream and
Lax-Wendroff schemes are also listed in Table 5.1. The result for the upstream
scheme is slightly in error, and could be improved by continuing the computa-