WENO Schemes
Weighted essentially non-oscillatory schemes achieve high order on smooth data while avoiding oscillations near discontinuities by adaptively weighting candidate stencils.
Choosing stencils by smoothness
Weighted essentially non-oscillatory (WENO) schemes reconstruct high-order interface values for finite-volume and finite-difference conservation laws by combining several candidate stencils. Each candidate spans a different set of cells; a smoothness indicator measures how oscillatory the solution is on each. The scheme then weights the candidates so that stencils crossing a discontinuity receive near-zero weight, while in smooth regions the weights combine to reach the highest possible order.
This adaptive weighting is the refinement over the earlier ENO idea, which simply picked the single smoothest stencil. WENO uses all of them convexly, gaining accuracy and smoothness of the weighting response.
Order and smoothness indicators
A fifth-order WENO scheme, common in practice, combines three third-order stencils. The smoothness indicators are sums of squared, scaled derivatives over each stencil; large values (steep or oscillatory data) drive that stencil's weight toward zero. In smooth regions the nonlinear weights approach fixed optimal weights that recover the full high order.
Strengths and costs
WENO captures shocks and steep gradients crisply without limiters, resolves smooth structures with high order, and handles complex wave interactions robustly. It is more expensive per point than a limited second-order scheme because of the multiple reconstructions and nonlinear weights, and its accuracy can degrade at critical points unless a mapped or improved weighting variant is used.
- Multiple candidate stencils combined by nonlinear weights
- Smoothness indicators suppress stencils near discontinuities
- Fifth-order from three third-order stencils is standard
- No explicit limiter needed for essentially non-oscillatory results
WENO reconstructions are widely used in high-fidelity compressible flow and MHD simulations where both sharp shocks and delicate smooth turbulence must be represented accurately at once.