282
5. Finite-Volume Methods
(Assuming that the CTU fluxes have been computed in the upstream direction,
these formulae apply regardless of the sign of the velocity.) The preceding corrections to the CTU fiux have exactly the same form as the corrections to the
upstream flux in the one-dimensional problem (5.41), which suggests that spurious oscillations in the vicinity of discontinuities or poorly resolved gradients can
be controlled if the corrections are limited using one of the flux limiter functions
discussed in Section 5.5.2. The resulting flux-limited approximation to the twodimensional advection problem is neither TVD nor monotone, but the spurious
oscillations generated by this scheme are extremely weak.
5.7.3 Nonuniform Nondivergent Flow
The generalization of this method to a nonuniform nondivergent velocity field
is most easily presented as the algorithm in Table 5.2, in which the fluxes are
initialized to zero at the beginning of each time step and then incrementally built
up in the course of two passes through the numerical mesh. The velocities are
assumed to be staggered such that ui+!,j and vi,i+! are displaced (!1s/2, 0) and
(0, !1s/2) away from the grid point where rPi,j is defined.?
If the flow is nondivergent, the algorithm in Table 5.2 can easily be recast as
an approximation to the transport equation in advective form (5.25) . When U and
v are positive, the upstream approximation to the spatial derivative operators in
(5.25) is
A • • _
u / ) - u · I ,
(rPi,j - rPi-J,i) + v . . I (rPi,j - rPi,i-J)
l ,j-1
.
,
/-1 ')
Sx
!1y
Assuming that the discretized velocity field satisfies the natural finite-difference
approximation to the nondivergence condition on a staggered mesh,
!1i,i may be expressed in the equivalent form
where FUP and GUp are the upstream fluxes defined in (5.52). The algorithm in
Table 5.2 may therefore be modified to yield a conservative advective-form approximation by replacing the three lines marked by stars with
9See Fig. 3.6 for an illustration of the same staggering scheme in a different context.
5. Finite-Volume Methods
(Assuming that the CTU fluxes have been computed in the upstream direction,
these formulae apply regardless of the sign of the velocity.) The preceding corrections to the CTU fiux have exactly the same form as the corrections to the
upstream flux in the one-dimensional problem (5.41), which suggests that spurious oscillations in the vicinity of discontinuities or poorly resolved gradients can
be controlled if the corrections are limited using one of the flux limiter functions
discussed in Section 5.5.2. The resulting flux-limited approximation to the twodimensional advection problem is neither TVD nor monotone, but the spurious
oscillations generated by this scheme are extremely weak.
5.7.3 Nonuniform Nondivergent Flow
The generalization of this method to a nonuniform nondivergent velocity field
is most easily presented as the algorithm in Table 5.2, in which the fluxes are
initialized to zero at the beginning of each time step and then incrementally built
up in the course of two passes through the numerical mesh. The velocities are
assumed to be staggered such that ui+!,j and vi,i+! are displaced (!1s/2, 0) and
(0, !1s/2) away from the grid point where rPi,j is defined.?
If the flow is nondivergent, the algorithm in Table 5.2 can easily be recast as
an approximation to the transport equation in advective form (5.25) . When U and
v are positive, the upstream approximation to the spatial derivative operators in
(5.25) is
A • • _
u / ) - u · I ,
(rPi,j - rPi-J,i) + v . . I (rPi,j - rPi,i-J)
l ,j-1
.
,
/-1 ')
Sx
!1y
Assuming that the discretized velocity field satisfies the natural finite-difference
approximation to the nondivergence condition on a staggered mesh,
!1i,i may be expressed in the equivalent form
where FUP and GUp are the upstream fluxes defined in (5.52). The algorithm in
Table 5.2 may therefore be modified to yield a conservative advective-form approximation by replacing the three lines marked by stars with
9See Fig. 3.6 for an illustration of the same staggering scheme in a different context.