278
5. Finite-Volume Methods
(1979) . Only the two -dimensional case will be considered here, for which a monotone low-order solution could be computed using the two-dimensional upstream
difference (3.31) . As discussed in Section 3.2.1, however, the CTU method (3.36)
is a better choice . The high -order fluxes could be estimated using an appropriate
form of the upstream biased Lax-Wendroff method (3.38). The generalization of
(3.38) to problems with spatially varying nondivergent winds will be discussed in
Section 5.7.3.
Suppose that i and j are the grid-point indices along the two spatial coordinates.
In contrast to the one-dimensional case, there will now be four antidiffusive fluxes
into each grid cell, and one must compute four coefficients
and
to
limit these fluxes. The formulae for C, '± I J' and Ci J'± I are identical to those given
2" '
'
2"
for C j±! in Section 5.4,2, except for the inclusion of the second dimension in the
subscript notation and the computation of the total antidiffusive fluxes in and out
of grid point i, j as
Pi'j = max (0,
+max (0,
- min (0,
- min (0,
+ max (0,
- min (0,
The formula for the permissible range of values for 4(jI also needs to be generalized to two dimens ions; the most natural choice is to define
= max
= min
,
and then let
= max
,
A,min
'f'i,j
.
= mm
(A,b A,b
A,b
A,b
A,b
'f'i,j' 'f'i,j-I' 'f'i,j+J' 'f'i-I, j' 'f'i+I,j
) .
The preceding technique for determining
and
does not completely prevent the development of small undershoots and overshoots in situations where
is being transported in a direction almost perpendicular to the gradient of Nevertheless, the spurious oscillations are typically very small and can be completely
eliminated using additional correction steps discussed by Zalesak.
5. Finite-Volume Methods
(1979) . Only the two -dimensional case will be considered here, for which a monotone low-order solution could be computed using the two-dimensional upstream
difference (3.31) . As discussed in Section 3.2.1, however, the CTU method (3.36)
is a better choice . The high -order fluxes could be estimated using an appropriate
form of the upstream biased Lax-Wendroff method (3.38). The generalization of
(3.38) to problems with spatially varying nondivergent winds will be discussed in
Section 5.7.3.
Suppose that i and j are the grid-point indices along the two spatial coordinates.
In contrast to the one-dimensional case, there will now be four antidiffusive fluxes
into each grid cell, and one must compute four coefficients
and
to
limit these fluxes. The formulae for C, '± I J' and Ci J'± I are identical to those given
2" '
'
2"
for C j±! in Section 5.4,2, except for the inclusion of the second dimension in the
subscript notation and the computation of the total antidiffusive fluxes in and out
of grid point i, j as
Pi'j = max (0,
+max (0,
- min (0,
- min (0,
+ max (0,
- min (0,
The formula for the permissible range of values for 4(jI also needs to be generalized to two dimens ions; the most natural choice is to define
= max
= min
,
and then let
= max
,
A,min
'f'i,j
.
= mm
(A,b A,b
A,b
A,b
A,b
'f'i,j' 'f'i,j-I' 'f'i,j+J' 'f'i-I, j' 'f'i+I,j
) .
The preceding technique for determining
and
does not completely prevent the development of small undershoots and overshoots in situations where
is being transported in a direction almost perpendicular to the gradient of Nevertheless, the spurious oscillations are typically very small and can be completely
eliminated using additional correction steps discussed by Zalesak.