  Embedded WENO methods utilize all adjacent smooth substencils to construct a
desirable interpolation. Conventional WENO schemes under-use this possibility
close to large gradients or discontinuities. We develop a general approach for
constructing embedded versions of existing WENO schemes. Embedded methods based
on the WENO schemes of Jiang and Shu (J. Comput. Phys. 126 (1996)) and on the
WENO-Z scheme of Borges et al. (J. Comput. Phys. 227 (2008)) are explicitly
constructed. Several possible choices are presented that result in either
better spectral properties or a higher order of convergence for sufficiently
smooth solutions. However, these improvements carry over to discontinuous
solutions. The embedded methods are demonstrated to be indeed improvements over
their standard counterparts by several numerical examples. All the embedded
methods presented have no added computational effort compared to their standard
counterparts.

Numerical Analysis