An extending strategy based on TENO framework for hyperbolic conservation laws

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.