5.5 Flux-Limiter Methods
263
5.4.3 Iterative Flux Correction
Substantial improvements in the neighborhood of smooth well-resolved extrema
can, nevertheless, be achieved by using a better estimate for the low-order solution, One strategy for obtaining a better low-order solution is to reuse the standard
FCT solution in an iterative application of the flux-correction procedure (Schär
and Smolarkiewicz 1996). As a consequence of the flux-correction algorithm, the
standard FCT solution is free from spurious ripples and can serve as an improved
estimate for the "transported and diffused solution" in a second iteration . That
portion of the antidiffusive flux that was not applied in the first iteration is the
maximum antidiffusive flux available for application in the second iteration. Letting the tilde denote a quantity defined for use in the second iteration, the final
step of the first iteration becomes
,p) -td , = ,p) td , - - /11 ( c
A '+
c)
I - A , 1 ,
/1x
and the new antidiffusive flux becomes
) 2
)-2
A)'+ I = A)'+I - AC'+I.
2
2 ) 2
The antidiffusive flux is limited using precisely the same flux-correction algorithm
used in the first iteration, and the final estimate for ,pn+I is obtained using
,p) n+1 , ,p)
-td
,
=
- - /11 (-c A '+ -c)
A ,
1 -
/1x
) 2
I
)-2
•
This iteration can be very effective in improving the solution near well-resolved
extrema such as the crest of a sine wave, but it does not noticeably improve the
solution near a discontinuous step.
5.5 Flux-Limiter Methods
The strategy behind flux-limiter methods is similar to that underlying flux-corrected transport in that the numerical fluxes used in both methods are a weighted
sum of the fluxes computed by a monotone first-order scheme and a higher-order
method. In flux-limiter methods, however, the limiter that apportions the flux between the high - and low-order schemes is determined without actually computing
a low-order solution (,ptd). This limiter is expressed as a function ofthe local solution at the previous time level in a manner guaranteeing that the scheme generates
TVD approximations to the one-dimensional scalar conservation law (5.6) and
that the scheme is second-order accurate except in the vicinity of the extrema of
,p.
Flux limiter methods approximate (5.6) with a finite-difference scheme in the
conservation form (5.26) using the flux
,+ I + C),+
1
1,+
I ) ,
F),+ I = F
2 ) 2
1
I - F
2 ) 2 ) 2
263
5.4.3 Iterative Flux Correction
Substantial improvements in the neighborhood of smooth well-resolved extrema
can, nevertheless, be achieved by using a better estimate for the low-order solution, One strategy for obtaining a better low-order solution is to reuse the standard
FCT solution in an iterative application of the flux-correction procedure (Schär
and Smolarkiewicz 1996). As a consequence of the flux-correction algorithm, the
standard FCT solution is free from spurious ripples and can serve as an improved
estimate for the "transported and diffused solution" in a second iteration . That
portion of the antidiffusive flux that was not applied in the first iteration is the
maximum antidiffusive flux available for application in the second iteration. Letting the tilde denote a quantity defined for use in the second iteration, the final
step of the first iteration becomes
,p) -td , = ,p) td , - - /11 ( c
A '+
c)
I - A , 1 ,
/1x
and the new antidiffusive flux becomes
) 2
)-2
A)'+ I = A)'+I - AC'+I.
2
2 ) 2
The antidiffusive flux is limited using precisely the same flux-correction algorithm
used in the first iteration, and the final estimate for ,pn+I is obtained using
,p) n+1 , ,p)
-td
,
=
- - /11 (-c A '+ -c)
A ,
1 -
/1x
) 2
I
)-2
•
This iteration can be very effective in improving the solution near well-resolved
extrema such as the crest of a sine wave, but it does not noticeably improve the
solution near a discontinuous step.
5.5 Flux-Limiter Methods
The strategy behind flux-limiter methods is similar to that underlying flux-corrected transport in that the numerical fluxes used in both methods are a weighted
sum of the fluxes computed by a monotone first-order scheme and a higher-order
method. In flux-limiter methods, however, the limiter that apportions the flux between the high - and low-order schemes is determined without actually computing
a low-order solution (,ptd). This limiter is expressed as a function ofthe local solution at the previous time level in a manner guaranteeing that the scheme generates
TVD approximations to the one-dimensional scalar conservation law (5.6) and
that the scheme is second-order accurate except in the vicinity of the extrema of
,p.
Flux limiter methods approximate (5.6) with a finite-difference scheme in the
conservation form (5.26) using the flux
,+ I + C),+
1
1,+
I ) ,
F),+ I = F
2 ) 2
1
I - F
2 ) 2 ) 2