5.7 Two Spatial Dimensions
287
Some idea of the cffectivencss of thc flux limiter may be obtained by comparing LeVeque's two-dimensional advection scheme with a high-order linear finitedifference scheme in which the spatial derivatives are computed using Lele's
compact fourth-order scheme (2.85), tirne-differencing is third-order AdamsBashforth, and the shortest wavelengths are smoothed using a sixth-order filter
(2.76) with Y6.1.t = 0.001. Due to the more stringent stability constraint associated with this scheme, the time steps were one-half those used in the corresponding flux-limited simulation. The numerical solution at t = 5 obtained with this
mcthod is shown in Fig. 5.21. Except for the nontrivial negative concentrations
generated by the linear high-order scheme, the overall character of the solution is
surprisingly similar to that generated by the two-dimensional flux-limited scheme.
In particular, the linear high-order scheme produces a similar tracer distribution
and does a slightly better job preserving the maxima in the concentration ficld.
The relative amplitudes of the negative concentrations generated by each method
are more clearly indicated in Figs. 5.22a and b, which compare the preceding
numerically computed concentrations along the line y = 0.5 at t = 2.5 with a
third numerical solution obtained using very fine spatial resolution. The spurious
negative concentrations produced by the high-order linear scheme are clearly evident, but aside from these negative concentrations and slight differences in the
amplitude of the peak concentration, the solutions are very similar.
The solution computed using the two-dimensional flux-limited scheme is rather
sensitive to the form of the flux limiter. This sensitivity is illustrated in Fig. 5.22c,
which compares the results obtained using superbee and Van Leer (5.39) lirniters and a 0.01 spatial mesh. The spurious damping of the peak concentration
generated by the Van Leer limiter is much more pronounced than that produced
by the superbee limiter. The incrcase in the effective numerical diffusion associated with the Van Leer limiter can also be appreciated by comparing Figs. 5.20
and 5.23, which show the solutions at t = 5 computed using the superbee and
Van Leer limiters, respectively. Although the Van Leer limiter generates considerably more diffusion, it has the advantage of not producing the spurious negative
concentrations obtained using thc superbee limiter. Note, however, that the negative concentrations generated by the superbee limiter are too small to be visible
in Fig. 5.22.
The performance of a time-split method is shown in Fig. 5.24. The advective
transports parallel to the x- and y-coordinates are computed in separate fractional steps, each of which used a superbee limiter in the one-dimensional fluxlimitcd algorithm described in Section 5.5.3. Strang splitting was not used and
thc operators don 't commute, so this method is not fully second-order in time.
The one-dimensional fluxes were calculated by replacing c in (5.42) with either
u ,i . or v
1+ 2 '}
i I , although this treatment of the spatial variations in the veloc1'} +2
ity field does not yield a fully second-order Lax-Wendroff approximation to the
one-dimensional variable-wind-speed advection equation.
Since the scheme used in each fractional step is the one-dimensional equivalent of LeVeque's two-dimensional flux-limited method, the difference between
287
Some idea of the cffectivencss of thc flux limiter may be obtained by comparing LeVeque's two-dimensional advection scheme with a high-order linear finitedifference scheme in which the spatial derivatives are computed using Lele's
compact fourth-order scheme (2.85), tirne-differencing is third-order AdamsBashforth, and the shortest wavelengths are smoothed using a sixth-order filter
(2.76) with Y6.1.t = 0.001. Due to the more stringent stability constraint associated with this scheme, the time steps were one-half those used in the corresponding flux-limited simulation. The numerical solution at t = 5 obtained with this
mcthod is shown in Fig. 5.21. Except for the nontrivial negative concentrations
generated by the linear high-order scheme, the overall character of the solution is
surprisingly similar to that generated by the two-dimensional flux-limited scheme.
In particular, the linear high-order scheme produces a similar tracer distribution
and does a slightly better job preserving the maxima in the concentration ficld.
The relative amplitudes of the negative concentrations generated by each method
are more clearly indicated in Figs. 5.22a and b, which compare the preceding
numerically computed concentrations along the line y = 0.5 at t = 2.5 with a
third numerical solution obtained using very fine spatial resolution. The spurious
negative concentrations produced by the high-order linear scheme are clearly evident, but aside from these negative concentrations and slight differences in the
amplitude of the peak concentration, the solutions are very similar.
The solution computed using the two-dimensional flux-limited scheme is rather
sensitive to the form of the flux limiter. This sensitivity is illustrated in Fig. 5.22c,
which compares the results obtained using superbee and Van Leer (5.39) lirniters and a 0.01 spatial mesh. The spurious damping of the peak concentration
generated by the Van Leer limiter is much more pronounced than that produced
by the superbee limiter. The incrcase in the effective numerical diffusion associated with the Van Leer limiter can also be appreciated by comparing Figs. 5.20
and 5.23, which show the solutions at t = 5 computed using the superbee and
Van Leer limiters, respectively. Although the Van Leer limiter generates considerably more diffusion, it has the advantage of not producing the spurious negative
concentrations obtained using thc superbee limiter. Note, however, that the negative concentrations generated by the superbee limiter are too small to be visible
in Fig. 5.22.
The performance of a time-split method is shown in Fig. 5.24. The advective
transports parallel to the x- and y-coordinates are computed in separate fractional steps, each of which used a superbee limiter in the one-dimensional fluxlimitcd algorithm described in Section 5.5.3. Strang splitting was not used and
thc operators don 't commute, so this method is not fully second-order in time.
The one-dimensional fluxes were calculated by replacing c in (5.42) with either
u ,i . or v
1+ 2 '}
i I , although this treatment of the spatial variations in the veloc1'} +2
ity field does not yield a fully second-order Lax-Wendroff approximation to the
one-dimensional variable-wind-speed advection equation.
Since the scheme used in each fractional step is the one-dimensional equivalent of LeVeque's two-dimensional flux-limited method, the difference between