A Parameter-Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

Abstract

A parameter-free limiting technique is developed for high-order unstruc- tured-grid methods to capture discontinuities when solving hyperbolic conserva- tion laws. The technique is based on a troubled-cell approach, in which cells re- quiring limiting are first marked, and then a limiter is applied to these marked A parameter-free accuracy-preserving TVD marker based on the cell-averaged so- lutions and solution derivatives in a local stencil is compared to several other mark- ers in the literature in identifying troubled cells. This marker is shown to be reliable and efficient to consistently mark the discontinuities. Then a compact high- order hierarchical moment limiter is developed for arbitrary unstructured grids. The limiter preserves a degree p polynomial on an arbitrary mesh. As a result, the solution accuracy near smooth local extrema is preserved. Numerical results for the high-order spectral difference methods are provided to illustrate the accuracy, effectiveness, and robustness of the present limiting technique. AMS subject classifications: 65M70, 76M20, 76M22