fluxlimiter

A flux limiter (often written flux-limiter) is a numerical device used in finite-volume and finite-difference schemes for hyperbolic partial differential equations to combine high-order accuracy in smooth regions with stable, non-oscillatory behavior near discontinuities. Flux limiters are nonlinear functions that adjust a high-order flux or slope correction using information about local gradients, effectively reducing spurious oscillations (Gibbs phenomena) while preserving sharp features such as shocks and contact discontinuities.

In practice, a limiter multiplies a high-order correction by a factor phi(r) that depends on a ratio r of successive solution differences. Typical formulae interpolate between a low-order, monotone scheme and a higher-order scheme: the limiter returns values near one in smooth regions and near zero near steep gradients, so the method reverts locally to a stable first-order discretization. Common limiter functions include minmod, Superbee, van Leer, and the monotonized central (MC) limiter. Flux-limited schemes are widely used with MUSCL, Godunov-type, Roe, and other shock-capturing methods.

Flux limiters are closely associated with the concept of total variation diminishing (TVD) schemes, which aim to prevent growth of total variation and thus avoid new extrema. The use of limiters represents a trade-off between accuracy and robustness: more compressive limiters preserve sharper discontinuities but can allow small overshoots, while more diffusive limiters are more robust but smear features. They are standard tools in computational fluid dynamics, astrophysical simulations, and other fields involving conservation laws.