Flux-Corrected Transport
A shock-capturing technique that blends a high-order flux with a diffusive low-order flux to prevent new extrema from forming.
The monotonicity dilemma
Advection of sharp features presents a classic conflict: high-order schemes are accurate but produce oscillations near discontinuities, while first-order upwind schemes are oscillation-free but smear features with excessive numerical diffusion. Flux-Corrected Transport (FCT), introduced by Boris and Book, was one of the earliest methods to get both accuracy and monotonicity by limiting fluxes locally.
The two-step recipe
FCT computes two fluxes at each interface: a high-order flux and a low-order (diffusive) flux. The low-order solution is guaranteed monotone but overly smeared. FCT then adds back as much of the difference between the high and low fluxes (the antidiffusive flux) as it safely can, limiting each antidiffusive flux so that no new maxima or minima are created and no existing extrema are amplified.
- Step 1: advance with the low-order monotone scheme
- Step 2: compute antidiffusive fluxes as the high-minus-low difference
- Step 3: limit each antidiffusive flux to enforce no new extrema
- Step 4: apply the limited antidiffusive fluxes to sharpen the solution
The flux limiter
The heart of FCT is the limiter, which examines neighboring values to decide how much antidiffusion is admissible. Zalesak's multidimensional limiter is the classical choice, computing bounds from local extrema and scaling fluxes to respect them. The limiter is what makes the scheme nonlinear even for linear advection, which is unavoidable for any accurate monotone method.
Relation to other methods
FCT is one member of the high-resolution family that includes total-variation-diminishing schemes and WENO. It remains popular in plasma fluid and MHD codes because it is robust, conservative, and generalizes cleanly to unstructured meshes and finite-element settings, where it is used to stabilize convection-dominated transport.