@article{xia_luo_frisbey_nourgaliev_2014, title={A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids}, volume={98}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2014.01.023}, abstractNote={A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches — analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.}, journal={COMPUTERS & FLUIDS}, author={Xia, Yidong and Luo, Hong and Frisbey, Megan and Nourgaliev, Robert}, year={2014}, month={Jul}, pages={134–151} }
 @article{xia_luo_nourgaliev_2014, title={An implicit Hermite WENO reconstruction-based discontinuous Galerkin method on tetrahedral grids}, volume={96}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2014.02.027}, abstractNote={An Implicit Reconstructed Discontinuous Galerkin method, IRDG (P1P2), is presented for solving the compressible Euler equations on tetrahedral grids. In this method, a quadratic polynomial (P2) solution is first reconstructed using a least-squares method from the underlying linear polynomial (P1) DG solution. By taking advantage of 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 and consistent with the underlying DG method. The final P2 solution is then obtained using a WENO reconstruction, which is necessary to ensure stability of the RDG (P1P2) method. A matrix-free GMRES (generalized minimum residual) algorithm is presented to solve the approximate system of linear equations arising from Newton linearization. The LU-SGS (lower–upper symmetric Gauss–Seidel) preconditioner is applied with both the simplified and approximate Jacobian matrices. The numerical experiments on a variety of flow problems demonstrate that the developed IRDG (P1P2) method is able to obtain a speedup of at least two orders of magnitude than its explicit counterpart, maintain the linear stability, and achieve the designed third order of accuracy: one order of accuracy higher than the underlying second-order DG (P1) method without significant increase in computing costs and storage requirements. It is also found that a well approximated Jacobian matrix is essential for the IRDG method to achieve fast converging speed and maintain robustness on large-scale problems.}, journal={COMPUTERS & FLUIDS}, author={Xia, Yidong and Luo, Hong and Nourgaliev, Robert}, year={2014}, month={Jun}, pages={406–421} }
 @article{luo_xia_li_nourgaliev_cai_2012, title={A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids}, volume={231}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2012.05.011}, abstractNote={A Hermite WENO reconstruction-based discontinuous Galerkin method RDG(P1P2), designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure linear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this RDG(P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a least-squares method from the underlying linear polynomial (P1) discontinuous Galerkin solution. 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 and consistent with the underlying DG method. The final quadratic polynomial solution is then obtained using a WENO reconstruction, which is necessary to ensure linear stability of the RDG method. The developed RDG method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments demonstrate that the developed RDG(P1P2) method is able to maintain the linear stability, achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method without significant increase in computing costs and storage requirements.}, number={16}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, Hong and Xia, Yidong and Li, Shujie and Nourgaliev, Robert and Cai, Chunpei}, year={2012}, month={Jun}, pages={5489–5503} }
 @article{luo_xia_spiegel_nourgaliev_jiang_2013, title={A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids}, volume={236}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2012.11.026}, DOI={10.1016/j.jcp.2012.11.026}, abstractNote={A reconstructed discontinuous Galerkin (RDG) method based on a hierarchical WENO reconstruction, termed HWENO (P1P2) in this paper, designed not only to enhance the accuracy of discontinuous Galerkin methods but also to ensure the nonlinear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this HWENO (P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a Hermite 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, thus ensuring the nonlinear stability of the RDG method. The developed HWENO (P1P2) method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, robustness, and non-oscillatory property. The numerical experiments indicate that the HWENO (P1P2) method is able to capture shock waves within one cell without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Luo, Hong and Xia, Yidong and Spiegel, Seth and Nourgaliev, Robert and Jiang, Zonglin}, year={2013}, month={Mar}, pages={477–492} }