Flux Limiters
Flux limiters blend high- and low-order schemes locally to keep sharp gradients monotone without spurious oscillations.
The accuracy-versus-oscillation dilemma
High-order schemes for advection are accurate on smooth solutions but produce overshoots and undershoots near steep gradients or discontinuities, the numerical analog of ringing. Low-order upwind schemes are oscillation-free but smear sharp features with excessive numerical diffusion. Flux limiters resolve this dilemma by switching adaptively: high order where the solution is smooth, low order where it is steep.
The switch is driven by a smoothness measure, usually the ratio r of consecutive solution gradients. A limiter function phi(r) multiplies the anti-diffusive part of the flux, turning it off (phi = 0) at extrema and on (approaching high order) in smooth regions.
The TVD constraint
Limiter functions are designed to keep the scheme total variation diminishing (TVD), which guarantees no new maxima or minima are created. Sweby's diagram defines the admissible region in the (r, phi) plane that a second-order TVD limiter must occupy. Common choices, minmod (most diffusive), van Leer, MC, and superbee (least diffusive, sharpest), trade smearing against steepness within this region.
Practical behavior
Minmod is safe but blunt; superbee keeps discontinuities crisp but can artificially steepen smooth gradients into false fronts. Van Leer and MC are popular compromises. The choice is problem dependent and affects how faithfully steep transport fronts are represented.
- Gradient ratio r measures local smoothness
- Limiter phi(r) blends high- and low-order flux
- Sweby region defines admissible second-order TVD limiters
- Minmod (diffusive) to superbee (compressive) spectrum
Flux limiters are standard in finite-volume transport and MHD codes wherever sharp fronts, such as edge-plasma gradients, must be advected without spurious oscillations corrupting the solution.