280
5. Finite-Volume Methods
In reality, the fluxes are transmitted parallel to the velocity vector, and an improved monotone scheme can be obtained by accounting for the transmission of
the fluxes at their correct angle to the coordinate axes, as illustrated in Fig. 5.17b.
The transport of these off-axis fluxes through each grid cell can be accounted for
by a modification of the basic upstream fluxes (5.52). Consider the corrections required to the flux through the lower boundary of cell Ci, j + 1). As highlighted by
the diagonal fill in Fig. 5.17c, the area swept out by the axis-parallel flux
I
"}+2
incorrectly includes a triangular region of cell Ci, j + 1) that is not penetrated by
the true advective flux though the lower cell boundary. The ratio of the area of
this triangular region to the area of the full grid cell is 0.5uv(ö,t)2 /(ö,s)2. There
is also a stippled triangular region of the same size that should receive a flux originating from cell (i - I, j). The axis-parallel upstream flux through the lower
boundary of cell Ci, j + 1) can be modified to account for these two corrections
accord ing to the formula
G ctu =
up
-
ö,t
2ö,s UV(
The upstream flux parallel to the x-axis must be similarly modified such that
ctu
up
ö,t
FO+ J 0= FO+ J ° - - 2UV( .
1 2 '}
1 2 '}
ö's
(5.54)
When the wind speed is constant, the method obtained by accounting for flux
propagation along the wind vector is identical to the CTU method (3.36). In Section 3.2.1 the CTU method was derived for a governing equation in advective form
using the method of characteristics to compute backward fluid-parcel trajectories.
An alternative derivation can be performed using the finite-volume formalism by
following trajectories backwards from the corner of each grid cell and then computing the average value of cell at the previous time step (Colella 1990). Using back trajectories to define the
subareas AI through A4 shown in Fig. 5.18, and assuming that the solution is
piecewise constant within each grid cell,
This finite-difference equation is not in conservation form, but it is equivalent to
the conservation form (5.51), with the fluxes given by (5.52), (5.53), and (5.54).
The CTU method will be monotone whenever the subareas AI through A4 are
positive, which in the general case where U and v can have arbitrary sign requires
that
ö,t
maxtja], lvI) ö's s 1.
(5.55)
The CTU method must be first-order accurate because it is a linear monotone scheme (Godunov 1959). As proposed by LeVeque (1996), an essentially
5. Finite-Volume Methods
In reality, the fluxes are transmitted parallel to the velocity vector, and an improved monotone scheme can be obtained by accounting for the transmission of
the fluxes at their correct angle to the coordinate axes, as illustrated in Fig. 5.17b.
The transport of these off-axis fluxes through each grid cell can be accounted for
by a modification of the basic upstream fluxes (5.52). Consider the corrections required to the flux through the lower boundary of cell Ci, j + 1). As highlighted by
the diagonal fill in Fig. 5.17c, the area swept out by the axis-parallel flux
I
"}+2
incorrectly includes a triangular region of cell Ci, j + 1) that is not penetrated by
the true advective flux though the lower cell boundary. The ratio of the area of
this triangular region to the area of the full grid cell is 0.5uv(ö,t)2 /(ö,s)2. There
is also a stippled triangular region of the same size that should receive a flux originating from cell (i - I, j). The axis-parallel upstream flux through the lower
boundary of cell Ci, j + 1) can be modified to account for these two corrections
accord ing to the formula
G ctu =
up
-
ö,t
2ö,s UV(
The upstream flux parallel to the x-axis must be similarly modified such that
ctu
up
ö,t
FO+ J 0= FO+ J ° - - 2UV(
1 2 '}
1 2 '}
ö's
(5.54)
When the wind speed is constant, the method obtained by accounting for flux
propagation along the wind vector is identical to the CTU method (3.36). In Section 3.2.1 the CTU method was derived for a governing equation in advective form
using the method of characteristics to compute backward fluid-parcel trajectories.
An alternative derivation can be performed using the finite-volume formalism by
following trajectories backwards from the corner of each grid cell and then computing the average value of cell at the previous time step (Colella 1990). Using back trajectories to define the
subareas AI through A4 shown in Fig. 5.18, and assuming that the solution is
piecewise constant within each grid cell,
This finite-difference equation is not in conservation form, but it is equivalent to
the conservation form (5.51), with the fluxes given by (5.52), (5.53), and (5.54).
The CTU method will be monotone whenever the subareas AI through A4 are
positive, which in the general case where U and v can have arbitrary sign requires
that
ö,t
maxtja], lvI) ö's s 1.
(5.55)
The CTU method must be first-order accurate because it is a linear monotone scheme (Godunov 1959). As proposed by LeVeque (1996), an essentially