On shock sensors for hybrid compact/WENO schemes

We present a systematic framework for the evaluation of shock sensors in high-resolution hybrid compact/WENO algorithms, with the goals of testing robustness and establishing accuracy in wavenumber space, with the intent to go beyond mere case-by-case comparison of solvers/sensors. Several sensors are considered, including classical ones, sensors based on multi-resolution wavelet analysis, and WENO-based shock sensors. The crucial issue of identification of suitable thresholds for the various shock sensors is tackled through a series of static tests and numerical simulations. Whereas any shock sensor with suitably low threshold is found to be effective in capturing shocks, not all of them perform equally well for waves. Performance degradation in numerical simulation of wave-like phenomena is here characterized by introducing an effective band-width for each sensor, which is found to be quite narrow for classical sensors, and much wider for WENO-based shock sensors. Based on this analysis, we also identify a new shock sensor which can be easily implemented in existing WENO-based codes. The new sensor doesn't suffer from contamination from the WENO numerical error even for waves resolved with as few as three points-per-wavelength. This property translates into improved performance in wavenumber space, and greater resolving power in flow cases involving shocks and compressible turbulence, as demonstrated through a series of numerical tests. We emphasize that this sensor can be applied more broadly to any algorithm which contains similar smoothness indicators as classical WENO.