A reconstructed discontinuous Galerkin method based on a hierarchical WENO reconstruction for computing shock waves on hybrid grids

A reconstructed discontinuous Galerkin (RDG) method based on a Hierarchical WENO reconstruction, termed HWENO(P₁P₂) in this paper, is developed for computing shock waves on 3D hybrid grids. The HWENO(P₁P₂) method is designed not only to enhance the accuracy of the discontinuous Galerkin (DG) method but also to ensure the nonlinear stability of the RDG method. In the HWENO(P₁P₂) method, a quadratic polynomial solution (P₂) is first reconstructed using a WENO reconstruction from the underlying linear polynomial (P₁) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities and thus ensure the nonlinear stability of the RDG method. The developed HWENO(P₁P₂) method is used to compute a variety of shock wave problems on 3D hybrid meshes to demonstrate its accuracy, robustness, versatility, and essentially non-oscillatory property. The numerical experiments indicate that the HWENO(P₁P₂) method is able to provide sharp resolution of discontinuities essentially without any spurious oscillations, and achieve the designed third-order of accuracy for smooth flows.