5.7 Two Spatial Dimensions
281
FIGURE 5.18. Backward trajectories from cell (i, j) defining the departure volume in the
CTU method.
second-order scheme can be obtained using the same strategy employed for the
one-dimensional flux-limiter methods : Corrective fluxes are added to Fctu and
G ctu that make the scheme equivalent to the Lax-Wendroff method except in regions where the solution is poorly resolved and the corrective flux is reduced to
minimize spurious oscillations. As discussed in Section 2.5.3, the Lax-Wendroff
approximation to the constant-wind-speed two-dimensional advection equation
has the form
where H (rP) is at least a first-order numerical approximation to
1/Itt = u
21/1x x + 2uv1/lxy + v
21/1y y,
and the subscripts on 1/1 denote partial derivatives . The divergence of the offaxis fluxes in the CTU method generate a first-order approximation to the mixed
partial derivative in the preceding. Thus, the only modifications that need to be
made to convert the CTU scheme to the Lax-Wendroff method are to replace the
one-sided approximations to 81/1/8x and 81/1/8y with centered second-order finite
differences and to include an approximation to
T Ilt ( 2
u 1/Ixx + v 2) 1/Iyy .
This can be accomplished by adding the following terms to the CTU fluxes:
Fi+U =
ctu
+ T lul ( 1 -Iul
Ilt)
Ils (rPi+l .j - rPi.j),
lvi (
(rPi
G, )"+1 = G " "+ 1 + -
ctu
I -Ivl- Ilt)
•
'1
I .) ' 1
2
)"+1 - rPi )").
.
I l s '