High-order multi-dimensional limiting strategy with subcell resolution I. Two-dimensional mixed meshes

The present paper deals with a new improvement of hierarchical multi-dimensional limiting process for resolving the subcell distribution of high-order methods on two-dimensional mixed meshes. From previous studies, the multi-dimensional limiting process (MLP) was hierarchically extended to the discontinuous Galerkin (DG) method and the flux reconstruction/correction procedure via reconstruction (FR/CPR) method on simplex meshes. It was reported that the hierarchical MLP (hMLP) shows several remarkable characteristics such as the preservation of the formal order-of-accuracy in smooth region and a sharp capturing of discontinuities in an efficient and accurate manner. At the same time, it was also surfaced that such characteristics are valid only on simplex meshes, and numerical Gibbs-Wilbraham oscillations are concealed in subcell distribution in the form of high-order polynomial modes. Subcell Gibbs-Wilbraham oscillations become potentially unstable near discontinuities and adversely affect numerical solutions in the sense of cell-averaged solutions as well as subcell distributions. In order to overcome the two issues, the behavior of the hMLP on mixed meshes is mathematically examined, and the simplex-decomposed P1-projected MLP condition and smooth extrema detector are derived. Secondly, a troubled-boundary detector is designed by analyzing the behavior of computed solutions across boundary-edges. Finally, hMLP_BD is proposed by combining the simplex-decomposed P1-projected MLP condition and smooth extrema detector with the troubled-boundary detector. Through extensive numerical tests, it is confirmed that the hMLP_BD scheme successfully eliminates subcell oscillations and provides reliable subcell distributions on two-dimensional triangular grids as well as mixed grids, while preserving the expected order-of-accuracy in smooth region.