A Face-Area-Weighted Centroid Formula for Reducing Grid Skewness and Improving Convergence of Edge-Based Solver on Highly-Skewed Simplex Grids

摘要

In this paper, we propose a dual-control-volume adjustment technique for improving the performance of edge-based discretization solvers on highly-skewed triangular grids. The idea is to reduce grid skewness and improve iterative flow-solver convergence by constructing the dual control volume with special element-centroids determined by a face-area-weighted centroid formula. In the resulting discretization, second-order accuracy is maintained on smooth grids but not on irregular grids, despite tremendous improvements in iterative convergence, because the altered centroids destroy the linear exactness property of the edge-based flux quadrature formula on such grids. As we will show, losing second-order accuracy has serious consequences on the quality of numerical solutions on irregular grids. In particular, the not-exactly-second-order method has been found to produce solutions contaminated with numerical noise. To address this issue, we propose to clean up the final solution with a post-processing operation based on an implicit second-order solver, in the same spirit as the filtering operation widely used in central-scheme-based large-eddy-simulations. Numerical results show that the dual-control-volume adjustment technique leads to superior iterative convergence for viscous flow problems on highly-skewed triangular grids and the post-processing effectively removes the numerical noise.