9 FLUX-CORRECTED TRANSPORT SCHEMES

Although flux-corrected transport (FCT) schemes have not enjoyed the popularity of total variation diminishing (TVD) schemes in the steady-state CFD community, they are included here for the following reasons.

(a) FCT schemes were the first to introduce the concept of limiters (Boris and Book (1973, 1976), Book and Boris (1975)), and particularly for didactic purposes it never hurts to go back to the source.

(b) FCT schemes are used almost exclusively in the unsteady CFD and plasma physics communities.

(c) Unlike TVD schemes, FCT limiting is based directly on the unknowns chosen. This makes FCT ideally suited to introduce the concept of nonlinear schemes and limiting.

FCT schemes were developed from the observation that linear schemes, however good, perform poorly for transient problems with steep gradients. For these linear schemes, the choice is between high-order, oscillatory, ‘noisy’ solutions, or low-order, overdiffusive, ‘smooth’ solutions. Godunov's theorem (Godunov (1959)) states this same observation as:

Theorem. (Godunov) No linear scheme of order greater than 1 will yield monotonic (wiggle-free, ripple-free) solutions.

The way to circumvent this stringent theorem is to develop nonlinear schemes. This is most easily accomplished by mixing a high-order and a low-order scheme. The process of combining these two schemes in a rational way is called limiting.

Any high-order scheme used to advance the solution either in time or between iterations ...

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