Improved eight-point targeted essentially non-oscillatory schemes using a three-point dynamic smooth stencil for compressible flows

In compressible flow, this work assumes that there is either a single discontinuity or two closely located discontinuities on a sufficiently refined mesh. Based on this assumption, a three-point dynamic smooth stencil with a large smoothness indicator is identified, which helps improve the smoothness detection for troubled stencils. This stencil satisfies the following properties: (1) Its smoothness indicator is the smallest among three non-overlapping three-point stencils. (2) It has the largest smoothness indicator among all stencils that satisfy condition (1). This large smoothness indicator is then used to rescale the cutoff threshold CT in the eight-point targeted essentially non-oscillatory schemes (Fu et al. in J Comput Phys 374:724–751, 2018). By incorporating this approach, the spectral properties of the linear scheme are recovered up to higher wavenumbers, and discontinuities are more effectively isolated from fluctuations. Numerical experiments demonstrate that the present method preserves the good shock-capturing capability of the original scheme, requires only a slight increase in computational time, and results in significant improvements in under-resolved turbulence statistical quantities.