A Discontinuous Galerkin Method Based on Variational Reconstruction for Compressible Flows on Arbitrary Grids

摘要

A reconstructed discontinuous Galerkin (rDG) method based on a variational formulation, is developed for compressible flows on 3D arbitrary grids. In this method, a high-order polynomial solution is reconstructed using a newly developed variational formulation based on the given linear discontinuous Galerkin (DG(P1)) solution. The high order terms are obtained by solving an extreme value problem, which minimizes the jump of the values of the reconstructed polynomial solutions and their spatial derivatives at cell interfaces, and therefore maximizes smoothness of the reconstructed polynomial solutions. The resulting method is stable even on tetrahedral grids, since its stencils are intrinsically the entire mesh. A variety of the benchmark test cases are presented to assess the accuracy, efficiency, robustness and flexibility of this rDG method. Hexahedral mesh, prismatic mesh and tetrahedral mesh arc tested in the presented work to illustrate that this method can be applied to arbitrary grids. The numerical experiments demonstrate that the developed method is able to maintain the linear stability, attain the designed high order of accuracy, and outperform the rDG method based on the least-squares reconstruction without a significant increase in computing costs and storage requirements.