5.5 Flux-LimiterMethods
267
required that the method be second-order accurate whenever r > O. Noting that
Cl : depends on the value of ifJj-2, the flux-limited scheme (5.34) has the form
All second-order approximations to the advection problem that have the preceding
form are weighted averages of the methods of Lax-Wendroff and of Warming
and Beam. As discussed previously, the flux-limited scheme becomes the LaxWendroff method if C (r) = 1. In a similar way, specifying C (r) = r converts the
scheme to the method of Warming and Beam (2.109). Curves corresponding to
are also plotted in Fig. 5.l1a. Of course, C LW (r) and CWB(r) do not lie entirely
within the shaded TVD region because neither the Lax-Wendroff method nor
that of Warming and Beam are TVD . Nevertheless, in order to make the ftuxlimited scheme second-order accurate away from local maxima and minima (i.e.,
for r > 0), C (r) must be a weighted average of C LW (r) and C WB (r). Sweby
suggests that the best results are obtained if this weighted average is an internal
average such that
C(r) = [1 - 9 (r)]C LW (r) + 9(r)C
W B(r),
(5.36)
where 0 S 9 (r) S 1. This portion of the total TVD region is indicated by the
shaded area in Fig. 5.11b, which will be referred to as the "second-order" TVD
region , although the true second-order TVD region includes extemal averages of
the Lax-Wendroff and Warming-Beam methods and is larger than the shaded area
in Fig. 5.11b.
5.5.2 Possible Flux Limiters
Possible choices for the specific functional form of C (r) that yield a TVD method
satisfying (5.36) include the "minrnod" limiter
C(r) = max[O, min(l, r)],
(5.37)
which is the dot-dashed curve following the lower boundary of the second-order
TVD region in Fig. 5.11b; the "superbee" limiter (Roe 1985)
C(r) = max[O, min(l, 2r), min(2, r)],
(5.38)
which lies along the upper boundary of the second-order TVD region; and the van
Leer limiter (van Leer 1974)
C r _ r + Irl
( ) - 1 + Irl '
(5.39)