A hybrid reconstructed discontinuous Galerkin method for compressible flows on arbitrary grids

A new reconstructed Discontinuous Galerkin (rDG) method based on a hybrid least-squares recovery and reconstruction, named P1P2(HLSr), is developed for solving the compressible Euler and Navier-Stokes equations on arbitrary grids. The development of the new hybrid rDG method is motivated by the observation that the original least-squares reconstruction does not have the property of the 2-exactness. As a remedy, the new hybrid reconstruction obtains a quadratic polynomial solution from the underlying linear DG solution by use of a hybrid recovery and reconstruction strategy. The resultant hybrid rDG method combines the simplicity of the reconstruction-based DG method and the accuracy of the recovery-based DG method, and has the desired property of 2-exactness. A number of test cases for a variety of flow problems are presented to assess the performance of the new P1P2(HLSr) method. Numerical experiments demonstrate that this hybrid rDG method is able to achieve the designed optimal 3rd order of accuracy for both inviscid and viscous flows and outperform the rDG methods based on either Green-Gauss or least-squares reconstruction.