Higher-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows

Abstract

This paper deals with the multi-dimensional limiting process (MLP) for discontinuous Galerkin (DG) methods to compute compressible inviscid and viscous flows. The MLP, which has been quite successful in finite volume methods (FVM), is extended to DG methods for hyperbolic conservation laws. In previous works, the MLP was shown to possess several superior characteristics, such as the ability to control multi-dimensional oscillation efficiently and to capture both discontinuous and continuous multi-dimensional flow features accurately within the finite volume framework. In particular, the oscillation-control mechanism in multiple dimensions was established by combining the local maximum principle and the multi-dimensional limiting (MLP) condition, leading to the formulation of efficient and accurate MLP-u slope limiters.The MLP limiting strategy is now extended to the higher-order DG framework to develop the hierarchical MLP formulation on unstructured grids, which facilitates the capturing of compressible flow structures very accurately. By combining the behavior of local extrema with the augmented MLP condition, the hierarchical MLP method (the MLP-based troubled-cell markers and the MLP limiters in the RKDG formulation) is developed for arbitrary DG-Pn polynomial approximations. Through extensive numerical analyses and computations on unstructured grids, it is demonstrated that the proposed hierarchical DG-MLP method yields outstanding performance in resolving non-compressive as well as compressive flow features.