A Parameter-Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids (Advances in Applied Mathematics and Mechanics, 2018)
A parameter-free limiting technique is developed for high-orderunstruc- tured-grid methods to capture discontinuities when solvinghyperbolic conservation laws. The technique is based on a"troubled-cell" approach, in which cells requiring limiting arefirst marked, and then a limiter is applied to these marked cells. Aparameter-free accuracy-preserving TVD marker based on thecell-averaged solutions and solution derivatives in a local stencilis compared to several other markers in the literature inidentifying "troubled cells". This marker is shown to be reliableand efficient to consistently mark the discontinuities. Then acompact high-order hierarchical moment limiter is developed forarbitrary unstructured grids. The limiter preserves a degree $p$ polynomial on an arbitrary mesh. As a result, the solution accuracynear smooth local extrema is preserved. Numerical results for thehigh-order spectral difference methods are provided to illustratethe accuracy, effectiveness, and robustness of the present limitingtechnique.