A Discontinuous Galerkin Method Based on a BGK Scheme for the Navier-Stokes Equations on Arbitrary Grids

A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook(BGK)formulation is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids.The idea behind this approach is to combine the robustness of the BGK scheme with the accuracy of the DG methods in an effort to develop a more accurate,efficient,and robust method for numerical simulations of viscous flows in a wide range of flow regimes.Unlike the traditional discontinuous Galerkin methods,where a Local Discontinuous Galerkin(LDG)formulation is usually used to discretize the viscous fluxes in the Navier-Stokes equations,this DG method uses a BGK scheme to compute the fluxes which not only couples the convective and dissipative terms together,but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function.The developed method is used to compute a variety of viscous flow problems on arbitrary grids.The numerical results obtained by this BGKDG method are extremely promising and encouraging in terms of both accuracy and robustness,indicating its ability and potential to become not just a competitive but simply a superior approach than the current available numerical methods.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

A reconstruction-based discontinuous Galerkin(RDG(P1P2))method,a variant of P1P2 method,is presented for the solution of the compressible Euler equations on arbitrary grids.In this method,an in-cell reconstruction,designed to enhance the accuracy of the discontinuous Galerkin method,is used to obtain a quadratic polynomial solution(P2)from the underlying linear polynomial(P1)discontinuous Galerkin solution using a least-squares method.The stencils used in the reconstruction involve only the von Neumann neighborhood(face-neighboring cells)and are compact and consistent with the underlying DG method.The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy,efficiency,robustness,and versatility.The numerical results indicate that this RDG(P1P2)method is third-order accurate,and outperforms the third-order DG method(DG(P2))in terms of both computing costs and storage requirements.

A Parallel,Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids

A reconstruction-based discontinuous Galerkin method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids.In this method,an in-cell reconstruction is used to obtain a higher-order polynomial representation of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction is used to obtain a continuous polynomial solution on the union of two neighboring,interface-sharing cells.The in-cell reconstruction is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution.The inter-cell reconstruction is devised to remove an interface discontinuity of the solution and its derivatives and thus to provide a simple,accurate,consistent,and robust approximation to the viscous and heat fluxes in the Navier-Stokes equations.A parallel strategy is also devised for the resulting reconstruction discontinuous Galerkin method,which is based on domain partitioning and Single Program Multiple Data(SPMD)parallel programming model.The RDG method is used to compute a variety of compressible flow problems on arbitrary meshes to demonstrate its accuracy,efficiency,robustness,and versatility.The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method,at the same time providing a better performance than the third order DG method,in terms of both computing costs and storage requirements.