WENOschemes

WENOschemes are a class of numerical methods used to solve hyperbolic conservation laws. They are designed to achieve high-order accuracy in smooth solution regions while suppressing spurious oscillations near discontinuities, such as shocks in fluid flows. The approach is based on reconstructing fluxes or variables from several candidate lower-order interpolations on a stencil and then combining them with nonlinear weights that adapt to local smoothness.

The construction relies on smoothness indicators that measure how rapidly the solution varies on each sub-stencil. In smooth regions, the nonlinear weights favor the highest-order reconstruction, yielding high accuracy. Near discontinuities, weights shift toward smoother stencils to prevent oscillations. The original framework was introduced by Jiang and Shu in 1996, and since then numerous variants have been developed, including WENO-JS and WENO-Z, which differ in how the smoothness indicators and weights are defined.

WENOschemes can be formulated in both finite-difference and finite-volume forms. They are typically used with flux-splitting or Riemann solvers and require reconstruction at cell interfaces to obtain numerical fluxes. Applications span one- and multi-dimensional problems in computational fluid dynamics, including the Euler equations, shallow-water equations, and other systems of hyperbolic balance laws.

Extensions and related methods include compact WENO (CWENO), Hermite WENO (HWENO), and adaptive or unstructured-grid versions, which extend the basic idea to different stencils, data representations, or grid geometries. The strengths of WENOschemes lie in their robustness near shocks and high-order accuracy in smooth regions, while their main limitations involve computational cost and implementation complexity for complex systems.