A third-order weighted variational reconstructed discontinuous Galerkin method for solving incompressible flows. (English) Zbl 1481.76073
Summary: In this paper, a third-order reconstructed discontinuous Galerkin (DG) method based on a weighted variational minimization principle, which is denoted as \(P_1 P_2\)(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 \(P_1 P_2\)(WVr) method. The numerical results demonstrate that the \(P_1 P_2\)(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 \(P_1 P_2\)(WVr) method outperforms the reconstructed DG methods based on either least-squares or Green-Gauss reconstruction for the simulations of incompressible flows.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
incompressible Navier-Stokes equations; reconstructed discontinuous Galerkin method; variational minimization principle; computational costsReferences:
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