Improving the quantification of overshooting shock-capturing oscillations

Abstract

A quantitative evaluation for shock-capturing over-amplification errors is provided to characterize the performance of shockcapturing schemes. Following our previous work on solely discontinuities, now we account for the concurrent presences of discontinuities and smooth waves, each with a complete set of supported modes on a given grid. The linear advection equation is taken as the model equation, and the standardized error evaluation is given for overshooting oscillations yielded by the shock-capturing schemes in the numerical solutions. Thus, we can quantitatively reveal the shock-capturing robustness of, for example, the well-known high-order WENO schemes, by comparing the resulting numerical overshoots. In particular, we are able to find out the ranges of wavenumbers in which the numerical schemes are especially prone to produce overshoots.While lower-dissipation is usually anticipated for high-order schemes, we provide a simple measure for quantifying the shockcapturing robustness, which was not a trivial task due to the nonlinearity of shock-capturing computations.