High-order embedded WENO schemes

\u3cp\u3eEmbedded WENO schemes are a new family of weighted essentially nonoscillatory schemes that always utilise all adjacent smooth substencils. This results in increased control over the convex combination of lower-order interpolations. We show that more conventional WENO schemes, such as WENO-JS and WENO-Z (Borges et al., J. Comput. Phys., 2008; Jiang and Shu, J. Comput. Phys., 1996), do not exhibit this feature and as such do not always provide a desirable linear combination of smooth substencils. In a previous work, we have already developed the theory and machinery needed to construct embedded WENO methods and shown some five-point schemes (van Lith et al., J. Comput. Phys., 2016). Here, we construct a seven-point scheme and show that it too performs well using some numerical examples from the one-dimensional Euler equations.\u3c/p\u3