Robust and Accurate Multi-dimensional Limiting Strategy for Higher-order Methods

Abstract

The present works deals with a robust and accurate multi-dimensional limiting strategy for higher-order CFD methods to analyze compressible flows. It consists of two parts: extension of multi-dimensional limiting process on unstructured grids and higher-order multi-dimensional limiting strategy. In first part, the multi-dimensional limiting process (MLP), which has been successfully proposed on structured grids in finite volume method (FVM), is extended to unstructured grids. The basic idea of the MLP limiting strategy is to control the distribution of both cell-averaged and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type linear reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured grids are obtained. Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the proposed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Examining robust multi-dimensional oscillation control mechanism, it is expected that MLP idea can be extended to higher-order CFD methods. Based on the MLP on FVM, the second part deals with extension of the MLP limiting philosophy into higher-order CFD methods. In order to enforce monotonicity for higher-order Pn proximation, two concepts are proposed: the augmented MLP condition and P1-projected MLP condition. Both conditions are successfully suppress multi-dimensional oscillations for arbitrary higher-order Pn approximation. Combining extrema detector, based on behavior of local smooth extrema, accurate and robust MLP based troubled-cell markers are developed. For the troubled-cells, the projection procedure and MLP slope limiter adjust sub-cell distributions. This limiting strategy are developed and implemented in the modal discontinuous Galerkin (DG) method and nodal correction procedure via reconstruction (CPR). Through extensive numerical analyses and computations on unstructured grids, it is demonstrated that the proposed limiting methods for higher-order CFD methods yields outstanding performance in resolving non-compressive as well as compressive flow features.