@article{li_pandare_luo_bakosi_waltz_2024, title={Robust 3D multi-material hydrodynamics using discontinuous Galerkin methods}, ISSN={["1097-0363"]}, DOI={10.1002/fld.5340}, abstractNote={Abstract A high‐order discontinuous Galerkin (DG) method is presented for nonequilibrium multi‐material () flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high‐order accurate solutions in smooth single‐material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high‐order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three‐dimensional multi‐material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi‐material solutions can be obtained by the DG method, as compared to second‐order finite volume methods.}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Li, Weizhao and Pandare, Aditya and Luo, Hong and Bakosi, Jozsef and Waltz, Jacob}, year={2024}, month={Oct} }
 @article{li_pandare_luo_bakosi_waltz_2023, title={A parallel p-adaptive discontinuous Galerkin method for the Euler equations with dynamic load-balancing on tetrahedral grids}, ISSN={["1097-0363"]}, DOI={10.1002/fld.5231}, abstractNote={AbstractA novel p‐adaptive discontinuous Galerkin (DG) method has been developed to solve the Euler equations on three‐dimensional tetrahedral grids. Hierarchical orthogonal basis functions are adopted for the DG spatial discretization while a third order TVD Runge‐Kutta method is used for the time integration. A vertex‐based limiter is applied to the numerical solution in order to eliminate oscillations in the high order method. An error indicator constructed from the solution of order  and  is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. The developed method is implemented with under the Charm++ parallel computing framework. Charm++ is a parallel computing framework that includes various load‐balancing strategies. Implementing the numerical solver under Charm++ system provides us with access to a suite of dynamic load balancing strategies. This can be efficiently used to alleviate the load imbalances created by p‐adaptation. A number of numerical experiments are performed to demonstrate both the numerical accuracy and parallel performance of the developed p‐adaptive DG method. It is observed that the unbalanced load distribution caused by the parallel p‐adaptive DG method can be alleviated by the dynamic load balancing from Charm++ system. Due to this, high performance gain can be achieved. For the testcases studied in the current work, the parallel performance gain ranged from 1.5× to 3.7×. Therefore, the developed p‐adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method without p‐adaptation.}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Li, Weizhao and Pandare, Aditya K. and Luo, Hong and Bakosi, Jozsef and Waltz, Jacob}, year={2023}, month={Aug} }
 @article{pandare_waltz_li_luo_bakosi_2023, title={On the design of stable, consistent, and conservative high-order methods for multi-material hydrodynamics}, volume={490}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2023.112313}, abstractNote={Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Pandare, Aditya K. and Waltz, Jacob and Li, Weizhao and Luo, Hong and Bakosi, Jozsef}, year={2023}, month={Oct} }
 @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_lohner_pandare_luo_2022, title={A vertex-centered finite volume method with interface sharpening technique for compressible two-phase flows}, volume={460}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2022.111194}, abstractNote={A robust and efficient finite volume method with interface sharpening technique has been developed to solve the six-equation multi-fluid single-pressure model for compressible two-phase flows. The numerical method is implemented in a three-dimensional vertex-centered code. A least-squares reconstruction with Kuzmin's vertex-based (VB) limiter is implemented for the volume fraction and a set of primitive variables in the presented finite volume framework. In regions where two different fluid components are present within a cell, a sharpening technique based on THINC (Tangent of Hyperbola for Interface Capturing) is adopted to provide a sharp resolution for the transitioning interface. These reconstructed values are then used as the initial data for Riemann problems. The enhanced AUSM+ -up scheme is applied to both liquid and gas flows. The multi-stage Runge-Kutta method is used for time marching. A number of benchmark test cases are presented to assess the performance of the present method. These include: an air-water interface moving at a constant velocity, Ransom's faucet problem, air-water/water-air shock tube problems with high pressure ratios, a shock in air impacting a water column case, an underwater explosion case and an air bubble blast case. In all of these cases, the shock and rarefaction waves are captured accurately, especially with the THINC interface sharpening technique. • The single pressure six-equation model is implemented in a three-dimensional vertex-centered code. • The Tangent of Hyperbola interface sharpening technique is implemented. • A number of test cases are presented to show the accuracy of the developed numerical method. • The numerical results in this paper are in good comparison with the literature.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, Lingquan and Lohner, Rainald and Pandare, Aditya K. and Luo, Hong}, year={2022}, month={Jul} }
 @article{luo_absillis_nourgaliev_2021, title={A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows}, volume={445}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2021.110618}, abstractNote={A moving discontinuous Galerkin finite element method with interface conservation enforcement (MDG+ICE) is developed for solving the compressible Euler equations. The MDG+ICE method is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a continuous space to determine discrete grid geometry. The resulting over-determined system of nonlinear equations arising from the MDG+ICE formulation can then be solved in a least-squares sense, leading to an unconstrained nonlinear least-squares problem that is regularized and solved by Levenberg-Marquardt method. A number of numerical experiments for compressible flows are conducted to assess the accuracy and robustness of the MDG+ICE method. Numerical results obtained indicate that the MDG+ICE method is able to implicitly detect and track all types of discontinuities via interface conservation enforcement and satisfy the conservation law on both elements and interfaces via grid movement and grid management, demonstrating that an exponential rate of convergence for Sod and Lax-Harden shock tube problems can be achieved and highly accurate solutions without overheating to both double-rarefaction wave and Noh problems can be obtained.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, Hong and Absillis, Gianni and Nourgaliev, Robert}, year={2021}, month={Nov} }
 @article{wang_luo_2021, title={A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids}, volume={3}, ISSN={["2524-6992"]}, DOI={10.1186/s42774-020-00055-6}, abstractNote={AbstractA high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.}, number={1}, journal={ADVANCES IN AERODYNAMICS}, author={Wang, Chuanjin and Luo, Hong}, year={2021}, month={Jan} }
 @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_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{chang_luo_2019, title={Second memorial issue in honor of Dr. Meng-Sing Liou}, volume={29}, ISSN={["1432-2153"]}, DOI={10.1007/s00193-019-00932-0}, number={8}, journal={SHOCK WAVES}, author={Chang, C. -H. and Luo, H.}, year={2019}, month={Nov}, pages={1007–1008} }
 @article{yang_cheng_luo_zhao_2018, title={A reconstructed direct discontinuous Galerkin method for simulating the compressible laminar and turbulent flows on hybrid grids}, volume={168}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2018.04.011}, abstractNote={A reconstructed direct discontinuous Galerkin method is developed and evaluated for simulating the compressible laminar and turbulent flows on hybrid grids. The spatial discretization is implemented through the combination of a highly efficient high-order hybrid reconstructed discontinuous Galerkin method together with a newly developed direct discontinuous Galerkin method for computing viscous fluxes. A variety of typical test cases are selected to evaluate the performance of the proposed hybrid reconstructed direct discontinuous Galerkin method for simulating both laminar flows and turbulent flows. Compared with the traditional discontinuous Galerkin method, the hybrid reconstructed direct discontinuous Galerkin method demonstrates its superior potential of being more accurate and more efficient under the same number of degrees of freedom, thus, shows its promise for further practical applications.}, journal={COMPUTERS & FLUIDS}, author={Yang, Xiaoquan and Cheng, Jian and Luo, Hong and Zhao, Qijun}, year={2018}, month={May}, pages={216–231} }
 @article{pandare_luo_2018, title={A robust and efficient finite volume method for compressible inviscid and viscous two-phase flows}, volume={371}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2018.05.018}, abstractNote={A robust and efficient density-based finite volume method is developed for solving the six-equation single pressure system of two-phase flows at all speeds on hybrid unstructured grids. Unlike conventional approaches where an expensive exact Riemann solver is normally required for computing numerical fluxes at the two-phase interfaces in addition to AUSM-type fluxes for single-phase interfaces in order to maintain stability and robustness in cases involving interactions of strong pressure and void-fraction discontinuities, a volume-fraction coupling term for the AUSM+-up fluxes is introduced in this work to impart the required robustness without the need of the exact Riemann solver. The resulting method is significantly less expensive in regions where otherwise the Riemann solver would be invoked. A transformation from conservative variables to primitive variables is presented and the primitive variables are then solved in the implicit method in order for the current finite volume method to be able to solve, effectively and efficiently, low Mach number flows in traditional multiphase applications, which otherwise is a great challenge for the standard density-based algorithms. A number of benchmark test cases are presented to assess the performance and robustness of the developed finite volume method for both inviscid and viscous two-phase flow problems. The numerical results indicate that the current density-based method provides an attractive and viable alternative to its pressure-based counterpart for compressible two-phase flows at all speeds.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Pandare, Aditya K. and Luo, Hong}, year={2018}, month={Oct}, pages={67–91} }
 @article{pandare_luo_bakosi_2019, title={An enhanced AUSM(+)-up scheme for high-speed compressible two-phase flows on hybrid grids}, volume={29}, ISSN={["1432-2153"]}, DOI={10.1007/s00193-018-0861-x}, abstractNote={An enhanced $$\hbox {AUSM}^+$$ -up scheme is presented for high-speed compressible two-phase flows using a six-equation two-fluid single-pressure model. Based on the observation that the $$\hbox {AUSM}^+$$ -up flux function does not take into account relative velocity between the two phases and thus is not stable and robust for computation of two-phase flows involving interaction of strong shock waves and material interfaces, the enhancement is in the form of a volume fraction coupling term and a modification of the velocity diffusion term, both proportional to the relative velocity between the two phases. These modifications in the flux function obviate the need to employ the exact Riemann solver, leading to a significantly less expensive yet robust flux scheme. Furthermore, the Tangent of Hyperbola for INterface Capturing (THINC) scheme is used in order to provide a sharp resolution for material interfaces. A number of benchmark test cases are presented to assess the performance and robustness of the enhanced $$\hbox {AUSM}^+$$ -up scheme for compressible two-phase flows on hybrid unstructured grids. The numerical experiments demonstrate that the enhanced $$\hbox {AUSM}^+$$ -up scheme along with THINC scheme can efficiently compute high-speed two-fluid flows such as shock–bubble interactions, while accurately capturing material interfaces.}, number={5}, journal={SHOCK WAVES}, author={Pandare, A. K. and Luo, H. and Bakosi, J.}, year={2019}, month={Jul}, pages={629–649} }
 @article{wu_shashkov_morgan_kuzmin_luo_2019, title={An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics}, volume={78}, ISSN={["1873-7668"]}, DOI={10.1016/j.camwa.2018.03.040}, abstractNote={We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy ( ρ , ρ u , E ) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energy ( ρ , u , e ). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.}, number={2}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Wu, T. and Shashkov, M. and Morgan, N. and Kuzmin, D. and Luo, H.}, year={2019}, month={Jul}, pages={258–273} }
 @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} }
 @article{xiaoquan_cheng_luo_zhao_2019, title={Robust Implicit Direct Discontinuous Galerkin Method for Simulating the Compressible Turbulent Flows}, volume={57}, ISSN={["1533-385X"]}, DOI={10.2514/1.J057172}, abstractNote={This paper proposes a robust and accurate implicit direct discontinuous Galerkin (DDG) method for simulating laminar and turbulent flows. The Reynolds averaged Navier–Stokes (RANS) equations couple...}, number={3}, journal={AIAA JOURNAL}, author={Xiaoquan, Yang and Cheng, Jian and Luo, Hong and Zhao, Qijun}, year={2019}, month={Mar}, pages={1113–1132} }
 @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{wang_cheng_berndt_carlson_luo_2018, title={Application of nonlinear Krylov acceleration to a reconstructed discontinuous Galerkin method for compressible flows}, volume={163}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2017.12.015}, abstractNote={A variant of Anderson Mixing, namely the Nonlinear Krylov Acceleration (NKA), is presented and implemented in a reconstructed Discontinuous Galerkin (rDG) method to solve the compressible Euler and Navier–Stokes equations on hybrid grids. A nonlinear system of equations as a result of a fully implicit temporal discretization at each time step is solved using the NKA method with a lower-upper symmetric Gauss–Seidel (LU-SGS) preconditioner. The developed NKA method is used to compute a variety of flow problems and compared with a well-known Newton-GMRES method to demonstrate the performance of the NKA method. Our numerical experiments indicate that the NKA method outperforms its Newton-GMRES counterpart for transient flow problems, and is comparable to Newton-GMRES for steady cases, and thus provides an attractive alternative to solve the system of nonlinear equations arising from the rDG approximation.}, journal={COMPUTERS & FLUIDS}, author={Wang, Chuanjin and Cheng, Jian and Berndt, Markus and Carlson, Neil N. and Luo, Hong}, year={2018}, month={Feb}, pages={32–49} }
 @article{lee_ahn_luo_2018, title={Cell-centered high-order hyperbolic finite volume method for diffusion equation on unstructured grids}, volume={355}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2017.10.051}, abstractNote={We apply a hyperbolic cell-centered finite volume method to solve a steady diffusion equation on unstructured meshes. This method, originally proposed by Nishikawa using a node-centered finite volume method, reformulates the elliptic nature of viscous fluxes into a set of augmented equations that makes the entire system hyperbolic. We introduce an efficient and accurate solution strategy for the cell-centered finite volume method. To obtain high-order accuracy for both solution and gradient variables, we use a successive order solution reconstruction: constant, linear, and quadratic (k-exact) reconstruction with an efficient reconstruction stencil, a so-called wrapping stencil. By the virtue of the cell-centered scheme, the source term evaluation was greatly simplified regardless of the solution order. For uniform schemes, we obtain the same order of accuracy, i.e., first, second, and third orders, for both the solution and its gradient variables. For hybrid schemes, recycling the gradient variable information for solution variable reconstruction makes one order of additional accuracy, i.e., second, third, and fourth orders, possible for the solution variable with less computational work than needed for uniform schemes. In general, the hyperbolic method can be an effective solution technique for diffusion problems, but instability is also observed for the discontinuous diffusion coefficient cases, which brings necessity for further investigation about the monotonicity preserving hyperbolic diffusion method.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Lee, Euntaek and Ahn, Hyung Taek and Luo, Hong}, year={2018}, month={Feb}, pages={464–491} }
 @inproceedings{liu_rollins_dinh_luo_2017, title={Sensitivity analysis of interfacial momentum closure terms in two phase flow and boiling simulations using MCFD solver}, volume={2}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85032902821&partnerID=MN8TOARS}, DOI={10.1115/ht2017-4963}, abstractNote={In this paper, a general workflow for the global Sensitivity Analysis (SA) has been proposed based on the coupling of VUQ toolkit DAKOTA and Multiphase Computational Fluid Dynamics (MCFD) solver boilEulerFoam. A surrogate model is first constructed based on sampling simulations from boilEulerFoam. This surrogate is based on Gaussian Processes Model (GPM) and is validated and proved to have good properties. The Morris Screening method is then applied based on the surrogate to those interfacial momentum closure terms for SA, including drag, lift, turbulent dispersion, wall lubrication, and virtual mass. Two different cases are considered, one is on low-pressure adiabatic flow, and the other is on high pressure boiling flow. Each case has its experimental background with data support. The radial void fraction distribution, gas velocity, relative velocity and liquid temperature (only for high pressure boiling case) are chosen as the Quantities of Interest (QoIs) which are of key interests for two-phase flow simulation and boiling crisis prediction. The interfacial force coefficient of each closure term is chosen as the input parameter. For the boiling case, the bubble diameter effect is also analyzed.
               Three remarks are drawn from this work on SA. First, it demonstrates the feasibility of surrogate model in the VUQ work for models in MCFD solver. The computational cost can be significantly reduced by employing the surrogate model. Secondly, through the Morris Sensitivity measurements, the importance of interfacial forces on different QoIs and regions can be analyzed and ranked for the two cases. Such analysis is also helpful for further model parameter calibration. Last but not least, the limitation of current work and the desired future work are discussed.}, booktitle={ASME 2017 Heat Transfer Summer Conference, HT 2017}, author={Liu, Y. and Rollins, C. and Dinh, Nam and Luo, H.}, year={2017} }
 @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{christon_bakosi_nadiga_berndt_francois_stagg_xia_luo_2016, title={A hybrid incremental projection method for thermal-hydraulics applications}, volume={317}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2016.04.061}, abstractNote={A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya–Babuška–Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie–Chow interpolation or by using a Petrov–Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with 2≤CFL≤100. The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e.g., in open-source packages such as HYPRE, PETSc, and Trilinos.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Christon, Mark A. and Bakosi, Jozsef and Nadiga, Balasubramanya T. and Berndt, Markus and Francois, Marianne M. and Stagg, Alan K. and Xia, Yidong and Luo, Hong}, year={2016}, month={Jul}, pages={382–404} }
 @article{pandare_luo_2016, title={A hybrid reconstructed discontinuous Galerkin and continuous Galerkin finite element method for incompressible flows on unstructured grids}, volume={322}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2016.07.002}, abstractNote={A hybrid reconstructed discontinuous Galerkin and continuous Galerkin method based on an incremental pressure projection formulation, termed rDG ( P n P m ) + CG ( P n ) in this paper, is developed for solving the unsteady incompressible Navier–Stokes equations on unstructured grids. In this method, a reconstructed discontinuous Galerkin method ( rDG ( P n P m ) ) is used to discretize the velocity and a standard continuous Galerkin method ( CG ( P n ) ) is used to approximate the pressure. The rDG ( P n P m ) + CG ( P n ) method is designed to increase the accuracy of the hybrid DG ( P n ) + CG ( P n ) method and yet still satisfy Ladyženskaja–Babuška–Brezzi (LBB) condition, thus avoiding the pressure checkerboard instability. An upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field. A number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rDG ( P n P m ) + CG ( P n ) method. The numerical experiments indicate that both rDG ( P 0 P 1 ) + CG ( P 1 ) and rDG ( P 1 P 2 ) + CG ( P 1 ) methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively. Moreover, the 3rd order rDG ( P 1 P 2 ) + CG ( P 1 ) method significantly outperforms its 2nd order rDG ( P 0 P 1 ) + CG ( P 1 ) and rDG ( P 1 P 1 ) + CG ( P 1 ) counterparts: being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Pandare, Aditya K. and Luo, Hong}, year={2016}, month={Oct}, pages={491–510} }
 @article{cheng_liu_luo_2016, title={A hybrid reconstructed discontinuous Galerkin method for compressible flows on arbitrary grids}, volume={139}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2016.04.001}, abstractNote={A new reconstructed Discontinuous Galerkin (rDG) method based on a hybrid least-squares recovery and reconstruction, named P1P2(HLSr), is developed for solving the compressible Euler and Navier-Stokes equations on arbitrary grids. The development of the new hybrid rDG method is motivated by the observation that the original least-squares reconstruction does not have the property of the 2-exactness. As a remedy, the new hybrid reconstruction obtains a quadratic polynomial solution from the underlying linear DG solution by use of a hybrid recovery and reconstruction strategy. The resultant hybrid rDG method combines the simplicity of the reconstruction-based DG method and the accuracy of the recovery-based DG method, and has the desired property of 2-exactness. A number of test cases for a variety of flow problems are presented to assess the performance of the new P1P2(HLSr) method. Numerical experiments demonstrate that this hybrid rDG method is able to achieve the designed optimal 3rd order of accuracy for both inviscid and viscous flows and outperform the rDG methods based on either Green-Gauss or least-squares reconstruction.}, journal={COMPUTERS & FLUIDS}, author={Cheng, Jian and Liu, Tiegang and Luo, Hong}, year={2016}, month={Nov}, pages={68–79} }
 @article{halashi_luo_2016, title={A reconstructed discontinuous Galerkin method for magnetohydrodynamics on arbitrary grids}, volume={326}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2016.08.055}, abstractNote={A reconstructed discontinuous Galerkin (rDG) method, designed not only to enhance the accuracy of DG methods but also to ensure the nonlinear stability of the rDG method, is developed for solving the Magnetohydrodynamics (MHD) equations on arbitrary grids. In this rDG(P1P2) method, a quadratic polynomial solution (P2) is first obtained using a Hermite Weighted Essentially Non-oscillatory (WENO) reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure linear stability of the rDG method and to improves efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the first derivatives in the DG formulation, the stencils used in 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 HLLD Riemann solver introduced in the literature for one-dimensional MHD problems is adopted in normal direction to compute numerical fluxes. The divergence free constraint is satisfied using the Locally Divergence Free (LDF) approach. The developed rDG method is used to compute a variety of 2D and 3D MHD problems on arbitrary grids to demonstrate its accuracy, robustness, and non-oscillatory property. Our numerical experiments indicate that the rDG(P1P2) method is able to capture shock waves sharply essentially 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}, author={Halashi, Behrouz Karami and Luo, Hong}, year={2016}, month={Dec}, pages={258–277} }
 @inbook{xia_frisbey_luo_2015, title={A Hierarchical WENO Reconstructed Discontinuous Galerkin Method for Computing Shock Waves}, ISBN={9783319168371 9783319168388}, url={http://dx.doi.org/10.1007/978-3-319-16838-8_25}, DOI={10.1007/978-3-319-16838-8_25}, booktitle={29th International Symposium on Shock Waves 2}, publisher={Springer International Publishing}, author={Xia, Y. and Frisbey, M. and Luo, H.}, year={2015}, pages={951–956} }
 @article{luo_edwards_luo_mueller_2015, title={A fine-grained block ILU scheme on regular structures for GPGPUs}, volume={119}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2015.07.005}, abstractNote={Iterative methods based on block incomplete LU (BILU) factorization are considered highly effective for solving large-scale block-sparse linear systems resulting from coupled PDE systems with n equations. However, efforts on porting implicit PDE solvers to massively parallel shared-memory heterogeneous architectures, such as general-purpose graphics processing units (GPGPUs), have largely avoided BILU, leaving their enormous performance potential unfulfilled in many applications where the use of implicit schemes and BILU-type preconditioners/solvers is highly preferred. Indeed, strong inherent data dependency and high memory bandwidth demanded by block matrix operations render naive adoptions of existing sequential BILU algorithms extremely inefficient on GPGPUs. In this study, we present a fine-grained BILU (FGBILU) scheme which is particularly effective on GPGPUs. A straightforward one-sweep wavefront ordering is employed to resolve data dependency. Granularity is substantially refined as block matrix operations are carried out in a true element-wise approach. Particularly, the inversion of diagonal blocks, a well-known bottleneck, is accomplished by a parallel in-place Gauss–Jordan elimination. As a result, FGBILU is able to offer low-overhead concurrent computation at O(n2N2) scale on a 3D PDE domain with a linear scale of N. FGBILU has been implemented with both OpenACC and CUDA and tested as a block-sparse linear solver on a structured 3D grid. While FGBILU remains mathematically identical to sequential global BILU, numerical experiments confirm its exceptional performance on an Nvidia GPGPU.}, journal={COMPUTERS & FLUIDS}, author={Luo, Lixiang and Edwards, Jack R. and Luo, Hong and Mueller, Frank}, year={2015}, month={Sep}, pages={149–161} }
 @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} }
 @article{xia_wang_luo_christon_bakosi_2016, title={Assessment of a hybrid finite element and finite volume code for turbulent incompressible flows}, volume={307}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.12.022}, abstractNote={Hydra-TH is a hybrid finite-element/finite-volume incompressible/low-Mach flow simulation code based on the Hydra multiphysics toolkit being developed and used for thermal-hydraulics applications. In the present work, a suite of verification and validation (V&V) test problems for Hydra-TH was defined to meet the design requirements of the Consortium for Advanced Simulation of Light Water Reactors (CASL). The intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of the Hydra-TH solution methods. The simulation problems vary in complexity from laminar to turbulent flows. A set of RANS and LES turbulence models were used in the simulation of four classical test problems. Numerical results obtained by Hydra-TH agreed well with either the available analytical solution or experimental data, indicating the verified and validated implementation of these turbulence models in Hydra-TH. Where possible, some form of solution verification has been attempted to identify sensitivities in the solution methods, and suggest best practices when using the Hydra-TH code.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Xia, Yidong and Wang, Chuanjin and Luo, Hong and Christon, Mark and Bakosi, Jozsef}, year={2016}, month={Feb}, pages={653–669} }
 @article{nourgaliev_luo_weston_anderson_schofield_dunn_delplanque_2016, title={Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change}, volume={305}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.11.004}, abstractNote={A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton–Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid–solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Nourgaliev, R. and Luo, H. and Weston, B. and Anderson, A. and Schofield, S. and Dunn, T. and Delplanque, J-R}, year={2016}, month={Jan}, pages={964–996} }
 @article{xia_lou_luo_edwards_mueller_2015, title={OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows}, volume={78}, ISSN={0271-2091}, url={http://dx.doi.org/10.1002/FLD.4009}, DOI={10.1002/FLD.4009}, abstractNote={SummaryAn OpenACC directive‐based graphics processing unit (GPU) parallel scheme is presented for solving the compressible Navier–Stokes equations on 3D hybrid unstructured grids with a third‐order reconstructed discontinuous Galerkin method. The developed scheme requires the minimum code intrusion and algorithm alteration for upgrading a legacy solver with the GPU computing capability at very little extra effort in programming, which leads to a unified and portable code development strategy. A face coloring algorithm is adopted to eliminate the memory contention because of the threading of internal and boundary face integrals. A number of flow problems are presented to verify the implementation of the developed scheme. Timing measurements were obtained by running the resulting GPU code on one Nvidia Tesla K20c GPU card (Nvidia Corporation, Santa Clara, CA, USA) and compared with those obtained by running the equivalent Message Passing Interface (MPI) parallel CPU code on a compute node (consisting of two AMD Opteron 6128 eight‐core CPUs (Advanced Micro Devices, Inc., Sunnyvale, CA, USA)). Speedup factors of up to 24× and 1.6× for the GPU code were achieved with respect to one and 16 CPU cores, respectively. The numerical results indicate that this OpenACC‐based parallel scheme is an effective and extensible approach to port unstructured high‐order CFD solvers to GPU computing. Copyright © 2015 John Wiley & Sons, Ltd.}, number={3}, journal={International Journal for Numerical Methods in Fluids}, publisher={Wiley}, author={Xia, Yidong and Lou, Jialin and Luo, Hong and Edwards, Jack and Mueller, Frank}, year={2015}, month={Feb}, pages={123–139} }
 @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_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={AbstractA 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} }
 @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} }
 @article{luo_segawa_visbal_2012, title={An implicit discontinuous Galerkin method for the unsteady compressible Navier-Stokes equations}, volume={53}, ISSN={["1879-0747"]}, DOI={10.1016/j.compfluid.2011.10.009}, abstractNote={A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier–Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p-multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady subsonic viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to significant improvements in performance over its explicit counterpart, while without significant increase in memory requirements.}, journal={COMPUTERS & FLUIDS}, author={Luo, Hong and Segawa, Hidehiro and Visbal, Miguel R.}, year={2012}, month={Jan}, pages={133–144} }
 @article{li_ren_lei_luo_2011, title={The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids}, volume={230}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2011.06.018}, abstractNote={Novel limiters based on the weighted average procedure are developed for finite volume methods solving multi-dimensional hyperbolic conservation laws on unstructured grids. The development of these limiters is inspired by the biased averaging procedure of Choi and Liu [10]. The remarkable features of the present limiters are the new biased functions and the weighted average procedure, which enable the present limiter to capture strong shock waves and achieve excellent convergence for steady state computations. The mechanism of the developed limiters for eliminating spurious oscillations in the vicinity of discontinuities is revealed by studying the asymptotic behavior of the limiters. Numerical experiments for a variety of test cases are presented to demonstrate the superior performance of the proposed limiters.}, number={21}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Li, Wanai and Ren, Yu-Xin and Lei, Guodong and Luo, Hong}, year={2011}, month={Sep}, pages={7775–7795} }
 @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}, abstractNote={Computational Fluid Dynamics Review 2010, pp. 103-122 (2010) No AccessA BGK-Based Discontinuous Galerkin Method for the Navier-Stokes Equations on Arbitrary GridsHong Luo, Luqing Luo and Kun XuHong LuoDepartment of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, 2414 Broughton Hall, Raleigh, NC, USA, Luqing LuoDepartment of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, 2414 Broughton Hall, Raleigh, NC, USA and Kun XuDepartment of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, Chinahttps://doi.org/10.1142/9789814313377_0006Cited by:1 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook (BGK) fonnulation 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) fonnulation 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 tenns 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 tenns 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. FiguresReferencesRelatedDetailsCited By 1High-order methods for diffuse-interface models in compressible multi-medium flows: A reviewV. Maltsev, M. Skote and P. Tsoutsanis1 Feb 2022 | Physics of Fluids, Vol. 34, No. 2 Computational Fluid Dynamics Review 2010Metrics History PDF download}, 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{ali_luo_syed_ishaq_2010, title={A Parallel Discontinuous Galerkin Code for Compressible Fluid Flows on Unstructured Grids}, volume={29}, number={1}, journal={Journal of Engineering and Applied Sciences}, author={Ali, A. and Luo, H. and Syed, K.S. and Ishaq, M.}, year={2010}, pages={57–75} }
 @article{luo_luo_ali_nourgaliev_cai_2011, title={A Parallel, Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids}, volume={9}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.070210.020610a}, abstractNote={AbstractA 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.}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Luo, Hong and Luo, Luqing and Ali, Amjad and Nourgaliev, Robert and Cai, Chunpei}, year={2011}, month={Feb}, pages={363–389} }
 @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} }
 @article{luo_spiegel_loehner_2010, title={Hybrid Grid Generation Method for Complex Geometries}, volume={48}, ISSN={["0001-1452"]}, DOI={10.2514/1.j050491}, abstractNote={A hybrid mesh generation method is described to discretize complex geometries. The idea behind this hybrid method is to combine the orthogonality and directionality of a structured grid, the efficiency and simplicity of a Cartesian grid, and the flexibility and ease of an unstructured grid in an attempt to develop an automatic, robust, and fast hybrid mesh generation method for configurations of engineering interest. A semistructured quadrilateral grid is first generated on the wetted surfaces. A background Cartesian grid, covering the domain of interest, is then constructed using a Quadtree-based Cartesian Method. Those Cartesian cells overlapping with the semistructured grids or locating outside of computational domain are then removed using an Alternating Digital Tree method. Finally, an unstructured grid generation method is used to generate unstructured triangular cells to fill all empty regions in the domain as a result of the trimming process. The automatic placement of sources at the geometrical irregularities is developed to render these regions isotropic, thus effectively overcoming the difficulty of generating highly stretched good-quality elements in these regions. The self-dividing of the exposed semistructured elements with high aspect ratio and the adaptation of the background mesh using the cell size information from the exposed semistructured elements for generating Cartesian cells are introduced to improve the quality of unstructured triangular elements and guarantee the success of the unstructured grid generation in the void regions. The developed hybrid grid generation method is used to generate a hybrid grid for a number of test cases, demonstrating its ability and robustness to mesh complex configurations.}, number={11}, journal={AIAA JOURNAL}, author={Luo, Hong and Spiegel, Seth and Loehner, Rainald}, year={2010}, month={Nov}, pages={2639–2647} }
 @article{luo_baum_lohner_2008, title={A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids}, volume={227}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2008.06.035}, abstractNote={A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest.}, number={20}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, Hong and Baum, Joseph D. and Lohner, Rainald}, year={2008}, month={Oct}, pages={8875–8893} }
 @article{luo_baum_loehner_2009, title={A hybrid building-block and gridless method for compressible flows}, volume={59}, ISSN={["1097-0363"]}, DOI={10.1002/fld.1827}, abstractNote={AbstractA hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.}, number={4}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Luo, Hong and Baum, Joseph D. and Loehner, Rainald}, year={2009}, month={Feb}, pages={459–474} }
 @article{luo_baum_lohner_2008, title={Fast p-multigrid discontinuous Galerkin method for compressible flows at all speeds}, volume={46}, ISSN={["1533-385X"]}, DOI={10.2514/1.28314}, abstractNote={Ap-multigrid (wherep is the polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature ofthisp-multigrid method is the application of an explicit smoother on the higher-level approximations (p > 0) and an implicit smoother on the lowest-level approximation (p = 0), resulting in a fast (and low) storage method that can be efficiently used to accelerate the convergence to a steady-state solution. Furthermore, this p-multigrid method can be naturally applied to compute the flows with discontinuities, in which a monotonic limiting procedure is usually required for discontinuous Galerkin methods. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries [Krivodonova, L., and Berger, M., "High-Order Implementation of Solid Wall Boundary Conditions in Curved Geometries," Journal of Computational Physics, Vol. 211, No.2,2006, pp. 492-512; and Luo, H., Baum, J. D., and Lohner, R., "On the Computation of Steady-State Compressible Flows Using a Discontinuous Galerkin Method," International Journal for Numerical Methods in Engineering, Vol. 73, No.5,2008, pp. 597-623]. A variety of compressible flow problems for a wide range of flow conditions from low Mach number to supersonic in both two-dimensional and three-dimensional configurations are computed to demonstrate the performance ofthisp-multigrid method. The numerical results obtained strongly indicate the order-independent property of this p-multigrid method and demonstrate that this method is orders-of-magnitude faster than its explicit counterpart The performance comparison with a finite volume method shows that using this p-multigrid method, the discontinuous Galerkin method, provides a viable, attractive, competitive, and probably even superior alternative to the finite volume method for computing compressible flows at all speeds.}, number={3}, journal={AIAA JOURNAL}, author={Luo, Hong and Baum, Joseph D. and Lohner, Rainald}, year={2008}, month={Mar}, pages={635–652} }
 @article{luo_baum_loehner_2007, title={A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids}, volume={225}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2006.12.017}, abstractNote={A weighted essentially non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for the discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and time-accurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, Hong and Baum, Joseph D. and Loehner, Rainald}, year={2007}, month={Jul}, pages={686–713} }
 @article{lohner_luo_baum_rice_2008, title={Improvements in speed for explicit, transient compressible flow solvers}, volume={56}, ISSN={["0271-2091"]}, DOI={10.1002/fld.1598}, abstractNote={AbstractSeveral explicit high‐resolution schemes for transient compressible flows with moving shocks are combined in such a way so as to achieve the highest possible speed without compromising accuracy. The main algorithmic changes considered comprise the following:

replacing limiting and approximate Riemann solvers by simpler schemes during the initial stages of Runge–Kutta solvers, and only using limiting and approximate Riemann solvers for the last stage;
automatically switching to simpler schemes for smooth flow regions;
automatic deactivation of quiescent regions; and
unstructured grids with cartesian cores or embedded cartesian grids.
The results from several examples demonstrate that speedup factors of 1:4 are attainable without compromising the accuracy of the traditional FCT schemes. Copyright © 2007 John Wiley & Sons, Ltd.}, number={12}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Lohner, Rainald and Luo, Hong and Baum, Joseph D. and Rice, Darren}, year={2008}, month={Apr}, pages={2229–2244} }
 @article{luo_baum_lohner_2008, title={On the computation of steady-state compressible flows using a discontinuous Galerkin method}, volume={73}, ISSN={["1097-0207"]}, DOI={10.1002/nme.2081}, abstractNote={AbstractComputation of compressible steady‐state flows using a high‐order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics‐based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low‐storage p‐multigrid method is used for solving the governing compressible Euler equations to obtain steady‐state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady‐state flows. Copyright © 2007 John Wiley & Sons, Ltd.}, number={5}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Luo, Hong and Baum, Joseph D. and Lohner, Rainald}, year={2008}, month={Jan}, pages={597–623} }
 @article{luo_baum_lohner_2006, title={A hybrid Cartesian grid and gridless method for compressible flows}, volume={214}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2005.10.002}, abstractNote={A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian grid is used as baseline mesh to cover the computational domain, while the boundary surfaces are addressed using a gridless method. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time-accurate solution of the compressible Euler equations. An overall speed-up factor of about eight and a saving in storage requirements about one order of magnitude for a typical three-dimensional problem in comparison with the unstructured grid method are demonstrated.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, H and Baum, JD and Lohner, R}, year={2006}, month={May}, pages={618–632} }
 @article{luo_baum_lohner_2006, title={A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids}, volume={211}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2005.06.019}, abstractNote={A p-multigrid (p = polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature of this p-multigrid method is to use different time integration schemes on different approximation levels, resulting in an accurate, fast, and low memory method that can be used to accelerate the convergence of the Euler equations to a steady state for discontinuous Galerkin methods. The developed method is used to compute the compressible flows for a variety of test problems on unstructured grids. The numerical results obtained strongly indicate the order independent property of this p-multigrid method. An overall speed-up factor more than one order of magnitude for both second- and third-order solutions of all test cases in comparison with the explicit method is demonstrated.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, H and Baum, JD and Lohner, R}, year={2006}, month={Jan}, pages={767–783} }
 @article{luo_baum_lohner_2005, title={Extension of Harten-Lax-van Leer scheme for flows at all speeds}, volume={43}, ISSN={["1533-385X"]}, DOI={10.2514/1.7567}, abstractNote={The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute flow problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range. ISTORICALLY, numerical algorithms for the solution of the Euler and Navier‐Stokes equations are classified as either pressure-based or density-based solution methods. The pressurebased methods, originally developed and well suited for incompressible flows, are typically based on the pressure correction techniques. They usually use a staggered grid and solve the governing equations in a segregated manner. The density-based methods, originally developed and robust for compressible flows, use time-arching procedures to solve the hyperbolic system of governing equations in a coupled manner. In general, density-based methods are not suitable for efficiently solving low Mach number or incompressible flow problems, because of large ratio of acoustic and convective timescales at the low-speed flow regimes. To alleviate this stiffness and associated convergence problems, time-derivative preconditioning techniques have been developed and used successfully for solving low-Machnumber and incompressible flows by many investigators, including Chorin, 1 Choi and Merkle, 2 Turkel, 3 Weiss and Smith, 4 and Dailey and Pletcher, 5 among others. Such methods seek to modify the time component of the governing equations so that the convergence can be made independent of Mach number. This is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order, and thus condition number remains bounded independent of the Mach number of the flows. Over the last two decades characteristic-based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible Euler equations. When these upwind methods are used to compute the numerical fluxes for the preconditioned Euler equations, solution accuracy at low speeds can be compromised, unless the numerical flux formulation is modified to take into account the eigensystem of the precondi}, number={6}, journal={AIAA JOURNAL}, author={Luo, H and Baum, JD and Lohner, R}, year={2005}, month={Jun}, pages={1160–1166} }
 @article{luo_baum_lohner_2005, title={High-Reynolds number viscous flow computations using an unstructured-grid method}, volume={42}, ISSN={["0021-8669"]}, DOI={10.2514/1.7575}, abstractNote={An unstructured grid method is presented to compute three-dimensional compressible turbulent flows for complex geometries. The Navier-Stokes equations, along with the one-equation turbulence model of Spalart-Allmaras are solved by the use of a parallel, matrix-free implicit method on unstructured tetrahedral grids. The developed method has been used to predict drags in the transonic regime for both DLR-F4 and DLR-F6 configurations to assess the accuracy and efficiency of the method}, number={2}, journal={JOURNAL OF AIRCRAFT}, author={Luo, H and Baum, JD and Lohner, R}, year={2005}, pages={483–492} }
 @article{löhner_luo_baum_2005, title={Selective edge removal for unstructured grids with Cartesian cores}, volume={206}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2004.11.034}, DOI={10.1016/j.jcp.2004.11.034}, abstractNote={Several rules for redistributing geometric edge-coefficient obtained for grids of linear elements derived from the subdivision of rectangles, cubes or prisms are presented. By redistributing the geometric edge-coefficient, no work is carried out on approximately half of all the edges of such grids. The redistribution rule for triangles obtained from rectangles is generalized to arbitrary situations in 3-D, and implemented in a typical 3-D edge-based flow solver. The results indicate that without degradation of accuracy, CPU requirements can be cut considerably for typical large-scale grids. This allows a seamless integration of unstructured grids near boundaries with efficient Cartesian grids in the core regions of the domain.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Löhner, Rainald and Luo, Hong and Baum, Joseph D.}, year={2005}, month={Jun}, pages={208–226} }
 @article{luo_baum_löhner_2003, title={Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids}, volume={17}, ISSN={1061-8562 1029-0257}, url={http://dx.doi.org/10.1080/1061856021000034337}, DOI={10.1080/1061856021000034337}, abstractNote={This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.}, number={1}, journal={International Journal of Computational Fluid Dynamics}, publisher={Informa UK Limited}, author={Luo, Hong and Baum, Joseph D. and Löhner, Rainald}, year={2003}, month={Jan}, pages={87–93} }
 @article{luo_baum_lohner_2004, title={On the computation of multi-material flows using ALE formulation}, volume={194}, ISSN={["0021-9991"]}, DOI={10.1016/j.jcp.2003.09.026}, abstractNote={Computation of compressible multi-fluid flows with a general equation of state using interface tracking and moving grid approach is discussed in this paper. The AUSM+, HLLC, and Godunov methods are presented and implemented in the context of arbitrary Lagrangian–Eulerian formulation for solving the unsteady compressible Euler equations. The developed methods are fully conservative, and used to compute a variety of multi-component flow problems, where the equations of state can be drastically different and stiff. Numerical results indicate that both ALE HLLC and Godunov schemes demonstrate their simplicity and robustness for solving such multi-phase flow problems, and yet ALE AUSM+ scheme exhibits strong oscillations around material interfaces even using a first order monotone scheme and therefore is not suitable for this class of problems.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, H and Baum, JD and Lohner, R}, year={2004}, month={Feb}, pages={304–328} }
 @article{sharov_luo_baum_lohner_2003, title={Unstructured Navier-Stokes grid generation at comers and ridges}, volume={43}, ISSN={["1097-0363"]}, DOI={10.1002/fld.615}, abstractNote={AbstractProblems related to automatic generation of highly stretched unstructured grids suitable for 3‐D Reynolds‐averaged Navier–Stokes computations are addressed. Special attention is given to treatment of such geometrical irregularities as convex and concave ridges as well as corners where the ridges meet. The existing unstructured grid generation approaches may fail or produce poor quality meshes in such geometrical regions. The proposed solution is based on special meshing of non‐slip body surfaces resulting in smooth and robust volume meshing and high overall quality of generated grids. Several examples demonstrate the efficiency of the method for complex 3‐D geometries. Copyright © 2003 John Wiley Sons, Ltd.}, number={6-7}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Sharov, D and Luo, H and Baum, JD and Lohner, R}, year={2003}, month={Oct}, pages={717–728} }
 @article{luo_baum_lohner_2001, title={An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids}, volume={30}, ISSN={["0045-7930"]}, DOI={10.1016/s0045-7930(00)00011-6}, abstractNote={Abstract An accurate, fast, matrix-free implicit method has been developed to solve the three-dimensional compressible unsteady flows on unstructured grids. A nonlinear system of equations as a result of a fully implicit temporal discretization is solved at each time step using a pseudo-time marching approach. A newly developed fast, matrix-free implicit method is then used to obtain the steady-state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady flow problems involving moving boundaries. The numerical results obtained indicate that the use of the present implicit method leads to a significant increase in performance over its explicit counterpart, while maintaining a similar memory requirement.}, number={2}, journal={COMPUTERS & FLUIDS}, author={Luo, H and Baum, JD and Lohner, R}, year={2001}, month={Feb}, pages={137–159} }
 @article{luo_sharov_baum_löhner_2001, title={On the Computation of Compressible Turbulent Flows on Unstructured Grids}, volume={14}, ISSN={1061-8562 1029-0257}, url={http://dx.doi.org/10.1080/10618560108940728}, DOI={10.1080/10618560108940728}, abstractNote={An accurate, fast, matrix-free implicit method has been developed to solve compressible turbulent How problems using the Spalart and Allmaras one equation turbulence model on unstructured meshes. The mean-flow and turbulence-model equations are decoupled in the time integration in order to facilitate the incorporation of different turbulence models and reduce memory requirements. Both mean flow and turbulent equations are integrated in time using a linearized implicit scheme. A recently developed, fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The spatial discretization is carried out using a hybrid finite volume and finite element method, where the finite volume approximation based on a containment dual control volume rather than the more popular median-dual control volume is used to discretize the inviscid fluxes, and the finite element approximation is used to evaluate the viscous flux terms. The developed method is used to compute a variety of turbulent flow problems in both 2D and 3D. The results obtained are in good agreement with theoretical and experimental data and indicate that the present method provides an accurate, fast, and robust algorithm for computing compressible turbulent flows on unstructured meshes.}, number={4}, journal={International Journal of Computational Fluid Dynamics}, publisher={Informa UK Limited}, author={Luo, Hong and Sharov, Dmitri and Baum, Joseph D. and Löhner, Rainald}, year={2001}, month={Jan}, pages={253–270} }
 @article{luo_2001, title={A fast, matrix-free implicit method for computing low Mach number flows on unstructured grids}, volume={14}, ISBN={NULL}, DOI={10.2514/6.1999-3315}, number={2}, journal={International Journal of Computational Fluid Dynamics}, author={Luo, H.}, year={2001}, pages={133–143} }
 @article{lohner_yang_baum_luo_pelessone_charman_1999, title={The numerical simulation of strongly unsteady flow with hundreds of moving bodies}, volume={31}, ISSN={["0271-2091"]}, DOI={10.1002/(sici)1097-0363(19990915)31:1<113::aid-fld958>3.0.co;2-q}, abstractNote={A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured grid, high-order, monotonicity preserving, ALE solver with automatic refinement and remeshing capabilities was enhanced by adding equations of state for high explosives, deactivation techniques and optimal data structures to minimize CPU overheads, automatic recovery of CAD data from discrete data, two new remeshing options, and a number of visualization tools for the preprocessing phase of large runs. The combination of these improvements has enabled the simulation of strongly unsteady flows with hundreds of moving bodies. Several examples demonstrate the effectiveness of the proposed methodology}, number={1}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Lohner, R and Yang, C and Baum, JD and Luo, H and Pelessone, D and Charman, CM}, year={1999}, month={Sep}, pages={113-+} }
 @article{luo_baum_lohner_1998, title={A fast, matrix-free implicit method for compressible flows on unstructured grids}, volume={146}, ISSN={["0021-9991"]}, DOI={10.1006/jcph.1998.6076}, abstractNote={A fast, matrix-free implicit method has been developed to solve the three-dimensional compressible Euler and Navier?Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower?upper symmetric Gauss?Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Luo, H and Baum, JD and Lohner, R}, year={1998}, month={Nov}, pages={664–690} }
 @inproceedings{lohner_yang_cebral_baum_mestreau_luo_pelessone_charman._1999, place={Trondheim}, title={Fluid-structure-thermal interaction using a loose coupling algorithm and adaptive unstructured grids}, DOI={10.2514/6.1998-2419}, booktitle={Computational methods for fluid-structure interaction : proceedings from the International Symposium on Computational Methods for Fluid-Structure Interaction, 15-17 February 1999}, publisher={Tapir Press}, author={Lohner, R. and Yang, C. and Cebral, J. and Baum, J.D. and Mestreau, E. and Luo, H. and Pelessone, D. and Charman., C.}, editor={Kvamsdal, TEditor}, year={1999}, pages={109–120} }
 @inbook{löhner_baum_yang_luo_1998, title={THE NUMERICAL SIMULATION OF STRONGLY UNSTEADY FLOW WITH HUNDREDS OF MOVING BODIES}, ISBN={9789810235642 9789812812957}, url={http://dx.doi.org/10.1142/9789812812957_0056}, DOI={10.1142/9789812812957_0056}, abstractNote={Computational Fluid Dynamics Review 1998, pp. 1027-1033 (1998) No AccessTHE NUMERICAL SIMULATION OF STRONGLY UNSTEADY FLOW WITH HUNDREDS OF MOVING BODIESRainald LÖHNER, Joseph D. BAUM, Chi YANG, and Hong LUORainald LÖHNERGMU/CSI, The George Mason University; Fairfax, VA 22030-4444, USA, Joseph D. BAUMScience Applications International Corporation;, 1710 Goodridge Drive, MS 2-3-1, McLean, VA 22102, USA, Chi YANGGMU/CSI, The George Mason University; Fairfax, VA 22030-4444, USA, and Hong LUOScience Applications International Corporation;, 1710 Goodridge Drive, MS 2-3-1, McLean, VA 22102, USAhttps://doi.org/10.1142/9789812812957_0056Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured-grid, high-order, monotonicity preserving, ALF solver with automatic refinement and remeshing capabilities was enhanced by adding: equations of state for high explosives, deactivation techniques and optimal data structures to minimize CPU overheads, two new remeshing options, and a number of visualization tools for the pre- processing phase of large runs. The combination of these improvements has enabled the simulation of strongly unsteady flows with hundreds of moving bodies. Several examples demonstrate the effectiveness of the proposed methodology. FiguresReferencesRelatedDetails Computational Fluid Dynamics Review 1998Metrics History PDF download}, booktitle={Computational Fluid Dynamics Review 1998}, publisher={WORLD SCIENTIFIC}, author={Löhner, Rainald and Baum, Joseph D. and Yang, Chi and Luo, Hong}, year={1998}, month={Nov}, pages={1027–1033} }
 @inbook{löhner_yang_cebral_baum_luo_pelessone_charman_1995, place={Chichester, England}, title={Fluid-Structure Interaction Using a Loose Coupling Algorithm and Adaptive Unstructured Grids}, ISBN={9780471955894}, booktitle={Computational fluid dynamics review 1995}, publisher={Wiley}, author={Löhner, R. and Yang, C. and Cebral, J. and Baum, J.D. and Luo, H. and Pelessone, D. and Charman, C.}, year={1995}, pages={755–776} }
 @article{luo_baum_lohner_1994, title={EDGE-BASED FINITE-ELEMENT SCHEME FOR THE EULER EQUATIONS}, volume={32}, ISSN={["0001-1452"]}, DOI={10.2514/3.12118}, abstractNote={We describe the development, validation, and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well-documented configurations. A flow solution about a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm}, number={6}, journal={AIAA JOURNAL}, author={LUO, H and BAUM, JD and LOHNER, R}, year={1994}, month={Jun}, pages={1183–1190} }