A reconstructed discontinuous Galerkin method for incompressible flows on arbitrary grids

Abstract

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific, the inviscid term of the INS equations is discretized by applying the third-order hybrid rDG(P1P2) method with a simplified artificial compressibility flux, while the viscous term of the INS equations is discretized by using the simple direct DG (DDG) method. A number of incompressible flow problems, in both steady and unsteady forms, for a variety flow conditions are computed to numerically assess the performance of the hybrid rDG(P1P2) method, which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs. Furthermore, a detailed comparison of a variety of different reconstructed strategies is performed and presented. Numerical results demonstrate that the hybrid rDG(P1P2) method outperforms the rDG(P1P2) method based on either the original least-squares reconstruction or the Green-Gauss reconstruction for solving the INS equations.