A third-order weighted variational reconstructed discontinuous Galerkin method for solving incompressible flows

Abstract

In this paper, a third-order reconstructed discontinuous Galerkin (DG) method based on a weighted variational minimization principle, which is denoted as P1P2(WVr) method, is presented for solving the incompressible flows on unstructured grids. In this method, the first-order degrees of freedom (DoFs) are obtained directly from the underlying second-order DG method, while the second-order DoFs are reconstructed through the weighted variational reconstruction. Specifically, we first introduce a weighted interfacial jump integration (WIJI) function which represents a measure of the jump between the reconstructed polynomial solutions from two neighboring cells. Then, we build the constitutive relations by minimizing this WIJI function using the variational method. A number of incompressible flow problems in both steady and unsteady forms are presented to assess the performance of the proposed P1P2(WVr) method. The numerical results demonstrate that the P1P2(WVr) method is able to achieve the designed optimal third-order accuracy at a significantly reduced computational costs. Moreover, when a suitable value of the weight parameter is chosen to be used, the P1P2(WVr) method outperforms the reconstructed DG methods based on either least-squares or Green-Gauss reconstruction for the simulations of incompressible flows.