Higher-Order Discontinuous Galerkin-MLP Methods on Triangular and Tetrahedral Grids

Abstract

This paper deals with the continuous e orts of extending the multi-dimensional limiting process (MLP) for high speed compressible ows (or hyperbolic conservation laws), which has been quite successful in nite volume methods (FVM), into discontinuous Galerkin (DG) methods. From the previous works, it was observed that the MLP shows several superior characteristics, such as e cient controlling of multi-dimensional oscillations and accurate capturing of both discontinuous and continuous multi-dimensional ow features within the nite volume framework. Especially, the fundamental mechanism of oscillation- control in multiple dimensions has been established by combining the local maximum prin- ciple and the multi-dimensional limiting (MLP) condition, which leads to the formulation of the e cient and accurate MLP-u slope limiters. The MLP limiting strategy is now extended into the higher-order DG framework to develop the hierarchical DG-MLP formulation of arbitrary higher-order (but up to P3 in present computations) on triangular and tetrahedral grids, which facilitates the capturing of detailed ow structures very accurately. By combining the conjecture on the behavior of local extrema with the augmented MLP condition, the hierarchical DG-MLP method (MLP-based trouble-cell markers and MLP limiters in RKDG formulation) are developed up to Pn. Through extensive numerical analyses and computations on triangular and tetra- hedral grids, it is demonstrated that the proposed DG-MLP approach yields outstanding performances in resolving non-compressive as well as compressive ow features.