A quantitative comparison of shock-capturing oscillations caused by high-order finite-difference schemes

Based on a quantitative evaluation process we have designed for assessing the robustness of shock/discontinuity-capturing finite-difference schemes, we compare the performance of various high-order schemes ranging from the classical weighted essentially non-oscillatory schemes to the novel targeted essentially non-oscillatory schemes. The results show that the schemes known for having better resolution may indeed produce stronger overshooting oscillations in various scenarios, where the CFL number, wave frequency, and distance between discontinuities vary. More importantly, the differences between the oscillations of different schemes are quantified systematically. Specifically, as the evaluation process includes a complete set of wave-interaction scenarios, we reveal that: (1) in certain ranges of wavenumber, a higher-order scheme may show smaller oscillations, and (2) in a certain range of CFL number, a smaller CFL number may lead to stronger oscillation. These two findings are difficult to observe while using the typical conventional evaluation process for robustness, which is in a case-by-case manner and without sufficient quantitative information. Potential usages of the present approach are also introduced.