ENO and WENO Schemes

The weighted essentially nonoscillatory (WENO) schemes, based on the successful essentially nonoscillatory (ENO) schemes with additional advantages, are a popular class of high-order accurate numerical methods for hyperbolic partial differential equations (PDEs) and other convection-dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high-order formal accuracy in smooth regions while maintaining stable, nonoscillatory and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution structures. In this chapter, we review the basic formulation of ENO and WENO schemes, outline the main ideas in constructing the schemes and discuss several of recent developments in using the schemes to solve hyperbolic type PDE problems.