5.4 Flux-Corrected Transport
261
5. Compute the required limitation on the net antidiffusive flux into grid point
j,
R-1: = Imin ( 1 , Qj/Pt )
J
0
if n- 0,
if Pt = o.
6. Compute the corresponding quantities involving the net antidiffusive flux
out of grid point i,
if P j
- > 0,
if Pj = O.
7. Limit the antidiffusive flux so that it neither produces an overshoot in the
grid cell into which it is directed nor generates an undershoot in the grid
cell out of which it flows:
if A i+ ! ::: 0,
if A i+ ! < o.
Two examples illustrating the performance of the Zalesak FCT algorithm on the
constant-wind-speed one-dimensional advection equation are shown in Fig. 5.10 .
In these examples, the monotone flux is computed using the upstream method
with
(5.29)
and the high-order flux is computed using the flux form of the Lax-Wendroff
method such that
h
c
c
2!:!.t
F 0
J+! = _(.I. 0+
2 'I'J
.I. 0 + J> - -(ifJ 0+1 _ .1.-).
'I'J
2!:!.x J
'I'J
(5.30)
The calculations were performed in a wide periodic domain, only the center portion of which is shown in each figure. In each case the wind speed is constant, and
the Courant number is 0.5.
The curves shown in Fig. 5.lOa are solutions to the same traveling-jump problem considered in connection with Fig. 5.7, except that the solutions are plotted at
t = 1.8. The solution computed using FCT is shown by the solid line. Also shown
are the exact solution and the approximate solutions obtained using upstream differencing and using the Lax-Wendroff method without FCT. The FCT scheme is
almost identical to the uncorrected Lax-Wendroff method except near the top of
the step, where the flux-correction procedure completely eliminates the dispersive