WENO and ENO Schemes
High-order finite-difference schemes for hyperbolic conservation laws that capture shocks sharply without spurious oscillations.
The problem with high order near shocks
Solutions to hyperbolic conservation laws (fluid flow, MHD) develop discontinuities such as shocks. A high-order polynomial reconstruction across a discontinuity produces Gibbs oscillations that pollute the solution and can trigger nonphysical values. Yet away from shocks, high order is desirable for accuracy. ENO and WENO schemes resolve this tension by adapting their stencil to the local smoothness of the data.
ENO: pick the smoothest stencil
The Essentially Non-Oscillatory (ENO) approach considers several candidate stencils for reconstructing a value at a cell interface and selects the one that avoids crossing a discontinuity, judged by divided differences. By always choosing the smoothest available stencil, ENO achieves high order in smooth regions while refusing to interpolate across shocks, keeping oscillations small.
WENO: blend all stencils
The Weighted ENO (WENO) improvement uses a convex combination of all candidate stencils rather than picking just one. Each stencil gets a nonlinear weight based on a smoothness indicator: smooth stencils get near-optimal weights, and stencils crossing a discontinuity are assigned near-zero weight. In smooth regions the weights combine to give a higher order of accuracy than any single stencil; near shocks the scheme automatically reverts to the safe, non-oscillatory stencils.
- Fifth-order WENO on a five-point stencil is a common workhorse
- Smoothness indicators measure the variation of the reconstruction polynomial on each stencil
- Paired with a strong-stability-preserving Runge-Kutta method in time
- Applied to characteristic variables for systems, to decouple wave families
Use in plasma and fluid codes
WENO schemes are standard in compressible fluid dynamics and are widely used for the ideal-MHD equations that model large-scale plasma dynamics, where sharp current sheets and shocks must be captured without oscillation. They compete with flux-corrected transport and slope-limited finite-volume methods for the same class of problems.