On the numerical overshoots of shock-capturing schemes

This note introduces a simple metric for benchmarking shock-capturing schemes, and this metric is especially focusing on the shock-capturing overshoots, which may damage the robustness of numerical simulations, as well as the reliability of numerical results. The idea is to numerically solve the linear advection equation with square waves of different wavenumbers as initial solutions. Therefore, the exact overshoots after one step of temporal evolution can be easily calculated, and shown as a function of CFL number and reduced wavenumber. The present metric is able to quantify the numerical overshoots of shock-capturing schemes, and some new features of several canonical schemes are shown.