a flux limiter
Numerical schemes for conservation laws face a cruel dilemma. Low-order (upwind, Godunov) schemes are rock-solid and oscillation-free but smear every sharp feature with numerical diffusion. High-order schemes are crisp and accurate in smooth regions but, near a discontinuity, overshoot and undershoot, throwing off spurious wiggles (Gibbs-like oscillations) that can even produce unphysical negative densities. A flux limiter is the clever device that gets the best of both: high accuracy where the solution is smooth, robust low-order behaviour exactly where it is sharp.
The mechanism is an adaptive blend, controlled by a local smoothness sensor. At each cell face you measure how smooth the solution is there, typically via the ratio r of consecutive gradients (the slope on one side divided by the slope on the other). A limiter function phi(r) then decides the recipe: where r indicates a smooth, monotone region, phi turns ON the high-order correction for accuracy; where r signals a steep gradient or a local extremum (a potential overshoot), phi smoothly dials the correction DOWN toward the safe low-order flux. The blended flux is the low-order flux plus phi(r) times the high-order correction. Popular limiters — minmod (most diffusive, safest), van Leer, superbee (sharpest, least diffusive), MC — are different choices of this phi, trading sharpness against robustness.
The theoretical anchor is the total variation diminishing (TVD) property: a good limiter is designed so the scheme's total variation (the sum of all the ups and downs of the solution) never increases in time, which is exactly the condition that forbids new oscillations from being created — no new wiggles, no overshoots. TVD schemes give physically sensible, monotonicity-preserving shock capture. The honest caveats: limiters are nonlinear (the only way past Godunov's first-order barrier without oscillation), they clip genuine smooth extrema down to first order (a real accuracy loss at peaks), and the strict TVD framework is essentially one-dimensional — multidimensional limiting along grid lines is a practical heuristic, not a theorem.
Advecting a profile that is a smooth bump next to a sharp step: with the minmod limiter the scheme runs at full second-order accuracy on the bump, but as it approaches the step the limiter detects the steep gradient and reverts to upwind, so the step stays monotone with no overshoot — exactly what a plain high-order scheme would fail to deliver.
High order on smooth bumps, low order at sharp steps — switched automatically.
TVD limiters provably suppress oscillations but they also CLIP smooth maxima and minima down to first-order accuracy — there is no free lunch. The flattened peak of a smooth pulse after long transport is the limiter doing its job, not a bug.