  Recently, the targeted ENO (TENO) schemes give a novel framework to keep
optimal high-order spatial reconstruction wherever discontinuity is deemed to
be vanished, including at smooth critical points, and to avoid oscillations by
completely removing stencils crossing discontinuities. Moreover, the smoothness
measurement of TENO schemes is in fact acting as shock-detectors, which are
capable for distinguishing discontinuities and smooth critical points.
Following the idea of a recent improvement, i.e. TENO-NA, the shock-detection
and stencil-selection are completely separated in this work. Higher-order
polynomials using neighbouring points of the standard five-point TENO scheme
are applied for achieving higher-order accuracy without significantly
increasing computation cost, by exploring the neighbouring smoothness
measurements. In this work, seventh-order spatial accuracy is achieved, and the
computational complexity is similar to that of the five-point TENO scheme.
Especially, the presented method introduces new flexibility in constructing
high-order numerical methods. Numerical results are given to show the
shock-capturing and wave-resolving capabilities.
