@article{luo_luo_nourgaliev_2012, title={A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids}, volume={12}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.250911.030212a}, abstractNote={Abstract 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.}, number={5}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Luo, Hong and Luo, Luqing and Nourgaliev, Robert}, year={2012}, month={Nov}, pages={1495–1519} }
@inbook{luo_luo_xu_2010, title={A BGK-Based Discontinuous Galerkin Method for the Navier-Stokes Equations on Arbitrary Grids}, volume={1}, ISBN={9789814313360 9789814313377}, url={http://dx.doi.org/10.1142/9789814313377_0006}, DOI={10.1142/9789814313377_0006}, number={3}, booktitle={Computational Fluid Dynamics Review 2010}, publisher={WORLD SCIENTIFIC}, author={Luo, Hong and Luo, Luqing and Xu, Kun}, year={2010}, month={Jul}, pages={103–122} }
@article{luo_luo_nourgaliev_mousseau_dinh_2010, title={A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids}, volume={229}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2010.05.033}, DOI={10.1016/j.jcp.2010.05.033}, abstractNote={A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier–Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. 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 method is able to deliver the same accuracy as the well-known Bassi–Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier–Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier–Stokes equations.}, number={19}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Luo, Hong and Luo, Luqing and Nourgaliev, Robert and Mousseau, Vincent A. and Dinh, Nam}, year={2010}, month={Sep}, pages={6961–6978} }