@article{li_liu_luo_2022, title={A reconstructed discontinuous Galerkin method based on variational formulation for compressible flows}, volume={466}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2022.111406}, abstractNote={A new reconstructed discontinuous Galerkin (rDG) method based on variational formulation is developed for compressible flows. In the presented method, a higher-order piece-wise polynomial is reconstructed based on the underlying discontinuous Galerkin (DG) solution. This reconstruction is done by using a newly developed variational formulation. The variational reconstruction (VR) can be seen as an extension of the compact finite difference (FD) schemes to unstructured grids. The reconstructed variables are obtained by solving an extreme-value problem, which minimizes the jumps of the reconstructed piece-wise polynomials across the cell interfaces, and therefore maximizes the smoothness of the reconstructed solution. Intrinsically, the stencils of the presented reconstruction are the entire mesh, so this method is robust even on tetrahedral grids. A variety of benchmark test cases are presented to assess the accuracy, efficiency and robustness of this rDG method. The numerical experiments demonstrate that the developed rDG method based on variational formulation can maintain the linear stability, obtain the designed high-order accuracy, and outperform the rDG counterpart based on the least-squares reconstruction for both inviscid and viscous compressible flows.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, Lingquan and Liu, Xiaodong and Luo, Hong}, year={2022}, month={Oct} }
 @article{li_lou_nishikawa_luo_2021, title={Reconstructed discontinuous Galerkin methods for compressible flows based on a new hyperbolic Navier-Stokes system}, volume={427}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2020.110058}, abstractNote={A new first-order hyperbolic system (FOHS) is formulated for the compressible Navier-Stokes equations. The resulting hyperbolic Navier-Stokes system (HNS), termed HNS20G in this paper, introduces the gradients of density, velocity, and temperature as auxiliary variables. Efficient, accurate, compact and robust reconstructed discontinuous Galerkin (rDG) methods are developed for solving this new HNS system. The newly introduced variables are recycled to obtain the gradients of the primary variables. The gradients of these gradient variables are reconstructed based on a newly developed variational formulation in order to obtain a higher order polynomial solution for these primary variables without increasing the number of degrees of freedom. The implicit backward Euler method is used to integrate solution in time for steady flow problems, while the third-order explicit first stage singly diagonally Runge-Kutta (ESDIRK) time marching method is implemented for advancing solutions in time for unsteady flows. The flux Jacobian matrices are obtained with an automatic differentiation toolkit TAPENADE. The approximate system of linear equations is solved with either symmetric Gauss-Seidel (SGS) method or general minimum residual (GMRES) algorithm with a lower-upper symmetric Gauss-Seidel (LU-SGS) preconditioner. A number of test cases are presented to assess accuracy and performance of the newly developed HNS+rDG methods for both steady and unsteady compressible viscous flows. Numerical experiments demonstrate that the developed HNS+rDG methods are able to achieve the designed order of accuracy for both primary variables and the their gradients, and provide an attractive and viable alternative for solving the compressible Navier-Stokes equations.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, Lingquan and Lou, Jialin and Nishikawa, Hiroaki and Luo, Hong}, year={2021}, month={Feb} }
 @article{lou_li_luo_nishikawa_2018, title={Reconstructed discontinuous Galerkin methods for linear advection-diffusion equations based on first-order hyperbolic system}, volume={369}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2018.04.058}, abstractNote={Newly developed reconstructed Discontinuous Galerkin (rDG) methods are presented for solving linear advection–diffusion equations on hybrid unstructured grids based on a first-order hyperbolic system (FOHS) formulation. Benefiting from both FOHS and rDG methods, the developed hyperbolic rDG methods are reliable, accurate, efficient, and robust, achieving higher orders of accuracy than conventional DG methods for the same number of degrees-of-freedom. Superior accuracy is achieved by reconstruction of higher-order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least-squares/variational reconstruction. Unsteady capability is demonstrated by an L-stable implicit time-integration scheme. A number of advection–diffusion test cases with a wide range of Reynolds numbers, including boundary layer type problems and unsteady cases, are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous grids, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the linear advection–diffusion equations.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Lou, Jialin and Li, Lingquan and Luo, Hong and Nishikawa, Hiroaki}, year={2018}, month={Sep}, pages={103–124} }