Improved Five- and Six-Point Targeted Essentially Nonoscillatory Schemes with Adaptive Dissipation

In this paper, we propose an improvement of the five-point and six-point targeted ENO schemes [Fu et al., JCP 305 (2016): 333-359] by introducing an adaptive dissipation control strategy. Nonlinear numerical dissipation is controlled by dynamically adjusting the cut-off parameter in the TENO weighting strategy according to the first-order smoothness measure of the local flow scales. For the five-point reconstruction, the dissipation bandwidth of the underlying linear scheme is delayed by introducing slight anti-dissipation at low wavenumbers with small dispersion errors. A new sixth order scale-separation parameter τ5 is derived and the modified TENO5-A scheme is third-order accurate. For the six-point reconstruction, dispersion and dissipation error are optimized separately resulting in a modified fourth-order TENO6-A scheme. All necessary parameters are determined by spectral analyses and are shown to be problem-independent by numerical experiments. A set of benchmark cases, including highly compressible gas dynamics, nearly incompressible and compressible turbulence decay, is considered. Numerical experiments demonstrate that both the proposed TENO5-A and TENO6-A scheme show excellent performance for shocks and broadband turbulence. The turbulence statistics obtained with TENO6-A at coarse resolution are comparable to those from the state-of-the-art implicit LES model and agree well with DNS data.