@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{lou_liu_luo_nishikawa_2019, title={Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids}, volume={25}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.OA-2017-0186}, abstractNote={Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic rDG methods can be made to have higher-order accuracy than conventional DG methods with fewer degrees of freedom. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. 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 girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations.}, number={5}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Lou, Jialin and Liu, Xiaodong and Luo, Hong and Nishikawa, Hiroaki}, year={2019}, month={May}, pages={1302–1327} }
 @article{cheng_liu_liu_luo_2017, title={A Parallel, High-Order Direct Discontinuous Galerkin Methods for the Navier-Stokes Equations on 3D Hybrid Grids}, volume={21}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.oa-2016-0090}, abstractNote={AbstractA parallel, high-order direct Discontinuous Galerkin (DDG) method has been developed for solving the three dimensional compressible Navier-Stokes equations on 3D hybrid grids. The most distinguishing and attractive feature of DDG method lies in its simplicity in formulation and efficiency in computational cost. The formulation of the DDG discretization for 3D Navier-Stokes equations is detailed studied and the definition of characteristic length is also carefully examined and evaluated based on 3D hybrid grids. Accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. The capability in handling curved boundary geometry is also demonstrated. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable results as the widely used BR2 scheme, clearly demonstrating that the DDG method provides an attractive alternative for solving the 3D compressible Navier-Stokes equations.}, number={5}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Cheng, Jian and Liu, Xiaodong and Liu, Tiegang and Luo, Hong}, year={2017}, month={May}, pages={1231–1257} }
 @article{liu_xia_luo_2018, title={A reconstructed discontinuous Galerkin method for compressible turbulent flows on 3D curved grids}, volume={160}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2017.10.014}, abstractNote={A third-order accurate reconstructed discontinuous Galerkin method, namely rDG(P1P2), is presented to solve the Reynolds-Averaged Navier–Stokes (RANS) equations, along with the modified one-equation model of Spalart and Allmaras (SA) on 3D curved grids. In this method, a piecewise quadratic polynomial solution (P2) is obtained using a least-squares method from the underlying piecewise linear DG(P1) solution. The reconstructed quadratic polynomial solution is then used for computing the inviscid and the viscous fluxes. Furthermore, Hermite Weighted Essentially Non-Oscillatory (WENO) reconstruction is used to guarantee the stability of the developed rDG method. A number of benchmark test cases based on a set of uniformly refined quadratic curved meshes are presented to assess the performance of the resultant rDG(P1P2) method for turbulent flow problems. The numerical results demonstrate that the rDG(P1P2) method is able to obtain reliable and accurate solutions to 3D compressible turbulent flows at a cost slightly higher than its underlying second-order DG(P1) method.}, journal={COMPUTERS & FLUIDS}, author={Liu, Xiaodong and Xia, Yidong and Luo, Hong}, year={2018}, month={Jan}, pages={26–41} }
 @article{liu_xuan_xia_luo_2017, title={A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on three-dimensional hybrid grids}, volume={152}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2017.04.027}, abstractNote={A reconstructed discontinuous Galerkin (rDG(P1P2)) method, originally introduced for the compressible Euler equations, is developed for the solution of the compressible Navier–Stokes equations on 3D hybrid grids. In this method, a piecewise quadratic polynomial solution is obtained from the underlying piecewise linear DG solution using a hierarchical Weighted Essentially Non-Oscillatory (WENO) reconstruction. The reconstructed quadratic polynomial solution is then used for the computation of the inviscid fluxes and the viscous fluxes using the second formulation of Bassi and Reay (Bassi–Rebay II). The developed rDG(P1P2) method is used to compute a variety of flow problems to assess its accuracy, efficiency, and robustness. The numerical results demonstrate that the rDG(P1P2) method is able to achieve the designed third-order of accuracy at a cost slightly higher than its underlying second-order DG method, outperform the third order DG method in terms of both computing costs and storage requirements, and obtain reliable and accurate solutions to the direct numerical simulation (DNS) of compressible turbulent flows.}, journal={COMPUTERS & FLUIDS}, author={Liu, Xiaodong and Xuan, Lijun and Xia, Yidong and Luo, Hong}, year={2017}, month={Jul}, pages={217–230} }
 @article{liu_xia_luo_xuan_2016, title={A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations}, volume={20}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.300715.140316a}, abstractNote={AbstractA comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.}, number={4}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Liu, Xiaodong and Xia, Yidong and Luo, Hong and Xuan, Lijun}, year={2016}, month={Oct}, pages={1016–1044} }
 @article{cheng_yang_liu_liu_luo_2016, title={A direct discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids}, volume={327}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2016.09.049}, abstractNote={A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Navier–Stokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the Navier–Stokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible Navier–Stokes equations. Based on our numerical results, we observe that DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible Navier–Stokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Cheng, Jian and Yang, Xiaoquan and Liu, Xiaodong and Liu, Tiegang and Luo, Hong}, year={2016}, month={Dec}, pages={484–502} }
 @article{xia_liu_luo_nourgaliev_2015, title={A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations}, volume={79}, ISSN={["1097-0363"]}, DOI={10.1002/fld.4057}, abstractNote={SummaryA space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.}, number={8}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Xia, Yidong and Liu, Xiaodong and Luo, Hong and Nourgaliev, Robert}, year={2015}, month={Nov}, pages={416–435} }