5.5 Flux-LimiterMethods
271
the same problem. The superiority of the MC flux-limited solution over the FCT
solution is not as clear-cutas in the examples shown in Fig. 5.13. The flux-limited
solution is more heavily damped but exhibits less phase -speed error than that obtained with flux-correct transport.
The performance of all the schemes shown in Figs . 5.12-5.14 is improved
by increasing the Courant number towards unity, since the upstream and LaxWendroff methods both give perfect results when JL = 1. In practical applications
with a temporally and spatially varying wind field there is, however, no hope of
stepping the solution forward at each grid point using a local Courant number
of unity. The solutions shown in Figs. 5.10-5.14 were obtained using JL = 0.5
and are similar to those obtained at smaller Courant numbers. Note that neither
the FCT nor the flux-limited methods approach the accuracy obtained on the test
problems in Figs. 5.13b and 5.14 using a simple explicit fourth-order spatial difference and an accurate time difference. Although FCT and flux-limiter methods
are highly useful in problems with discontinuities and unresolved gradients, conventional finite-difference schemes may perform much better when the solution is
at least moderately resolved. The resolution required to make a high-order finitedifference method attractive need not be particularly high; the 7.5ßx-wavelength
component in the solution shown in Fig. 5.14 is certainly not well-resolved,
5.5.3 Flow Velocities 01Arbitrary Sign
In order to accommodate velocities of arbitrary sign , the definitions of Fj+t and
Tj+t must be modified as follows. The Lax-Wendroff flux (5.30) may be expressed in terms of the upstream flux for advection by a velocity of arbitrary sign
(5.29) as
h
F j +
I
[c] (
ICIßt) ( n
n)
t
= Fj+t + 2 1 - ßX tPj+1 - tPj .
(5.41)
The total corrected flux may therefore be expressed as
n
C ( n n) 1 [
n
c
2 ßt n ] ( n n)
F j + t
= 2 tPj+1 + tPj - 2 Cl - Cj+t)lcl + ßx Cj+t tPj+1 - tPj .
(5.42)
The value of T J '+ I used in the evaluation of the flux limiter
1 should be
!
J+!
computed as ratio of the slope of the solution across the cell interface upstream
of j+t to the slope of the solution across the interface at j+t. Defining y =
-sgn(cj), this ratio becomes
n
_ tP J +y+1 - tP J +y
T '+ 1 -
n
n
J !
tPj+1 - tP j
If the velocity varies as a function of time, an 0 [(ßt)2] approximation to the
velocity at (n+t)ßt should be used in (5.42) to preserve second-order accuracy
(at least at locations away from the extrema of tP). Suitable approximations can
271
the same problem. The superiority of the MC flux-limited solution over the FCT
solution is not as clear-cutas in the examples shown in Fig. 5.13. The flux-limited
solution is more heavily damped but exhibits less phase -speed error than that obtained with flux-correct transport.
The performance of all the schemes shown in Figs . 5.12-5.14 is improved
by increasing the Courant number towards unity, since the upstream and LaxWendroff methods both give perfect results when JL = 1. In practical applications
with a temporally and spatially varying wind field there is, however, no hope of
stepping the solution forward at each grid point using a local Courant number
of unity. The solutions shown in Figs. 5.10-5.14 were obtained using JL = 0.5
and are similar to those obtained at smaller Courant numbers. Note that neither
the FCT nor the flux-limited methods approach the accuracy obtained on the test
problems in Figs. 5.13b and 5.14 using a simple explicit fourth-order spatial difference and an accurate time difference. Although FCT and flux-limiter methods
are highly useful in problems with discontinuities and unresolved gradients, conventional finite-difference schemes may perform much better when the solution is
at least moderately resolved. The resolution required to make a high-order finitedifference method attractive need not be particularly high; the 7.5ßx-wavelength
component in the solution shown in Fig. 5.14 is certainly not well-resolved,
5.5.3 Flow Velocities 01Arbitrary Sign
In order to accommodate velocities of arbitrary sign , the definitions of Fj+t and
Tj+t must be modified as follows. The Lax-Wendroff flux (5.30) may be expressed in terms of the upstream flux for advection by a velocity of arbitrary sign
(5.29) as
h
F j +
I
[c] (
ICIßt) ( n
n)
t
= Fj+t + 2 1 - ßX tPj+1 - tPj .
(5.41)
The total corrected flux may therefore be expressed as
n
C ( n n) 1 [
n
c
2 ßt n ] ( n n)
F j + t
= 2 tPj+1 + tPj - 2 Cl - Cj+t)lcl + ßx Cj+t tPj+1 - tPj .
(5.42)
The value of T J '+ I used in the evaluation of the flux limiter
1 should be
!
J+!
computed as ratio of the slope of the solution across the cell interface upstream
of j+t to the slope of the solution across the interface at j+t. Defining y =
-sgn(cj), this ratio becomes
n
_ tP J +y+1 - tP J +y
T '+ 1 -
n
n
J !
tPj+1 - tP j
If the velocity varies as a function of time, an 0 [(ßt)2] approximation to the
velocity at (n+t)ßt should be used in (5.42) to preserve second-order accuracy
(at least at locations away from the extrema of tP). Suitable approximations can