On the quantification of overshooting shock-capturing oscillations

An evaluation method for the shock-capturing over-amplification error (overshoot) is provided to quantitatively characterize the shock-capturing performance of nonlinear finite-difference schemes. In this evaluation method, the linear advection equation is taken as the model equation. We account for the concurrent presence of discontinuities and smooth waves, each with a complete set of supported modes. The quantitative error evaluation relies on analyzing overshoots yielded by shock-capturing schemes while solving the linear advection equation. TVD schemes and the ENO3 scheme are shown to be able to provide overshoot-free solutions, but certain schemes such as high-order WENO schemes can overshoot the exact solutions significantly in the vicinity of the discontinuities. More importantly, we can specifically and quantitatively find out the ranges of wavenumbers in which the numerical schemes are especially prone to produce overshoots. Although it is known that higher-order schemes are easier to produce overshoots, the present method quantifies the performance of shock-capturing schemes. Moreover, the present method is simple to use.