Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations
Year: 2013
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 1013–1044
Abstract
A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.020611.090312a
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 1013–1044
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
-
Numerical prediction of drag crisis for smooth spheres using a high-order flux reconstruction method
Akbarzadeh, Amir | Alhawwary, Mohammad | Witherden, Freddie | Jameson, AntonyAIAA SCITECH 2023 Forum, (2023),
https://doi.org/10.2514/6.2023-2146 [Citations: 0] -
Implementation of a FWH approach in a high-order LES tool for aeroacoustic noise predictions
Alhawwary, Mohammad A. | Wang, Zhi J.AIAA Scitech 2020 Forum, (2020),
https://doi.org/10.2514/6.2020-1724 [Citations: 8] -
An implicit high‐order preconditioned flux reconstruction method for low‐Mach‐number flow simulation with dynamic meshes
Wang, Lai | Yu, MeilinInternational Journal for Numerical Methods in Fluids, Vol. 91 (2019), Iss. 7 P.348
https://doi.org/10.1002/fld.4759 [Citations: 9] -
A Combined-Mode Fourier Analysis of DG Methods for Linear Parabolic Problems
Alhawwary, Mohammad | Wang, ZhijianSIAM Journal on Scientific Computing, Vol. 42 (2020), Iss. 6 P.A3825
https://doi.org/10.1137/20M1316962 [Citations: 7] -
A Parallel, High-Order Direct Discontinuous Galerkin Method for the Navier-Stokes Equations on 3D Hybrid Grids
Cheng, Jian | Liu, Xiaodong | Liu, Tiegang | Luo, HongCommunications in Computational Physics, Vol. 21 (2017), Iss. 5 P.1231
https://doi.org/10.4208/cicp.OA-2016-0090 [Citations: 32] -
Higher-order multi-dimensional limiting process for DG and FR/CPR methods on tetrahedral meshes
Park, Jin Seok | You, Hojun | Kim, ChongamComputers & Fluids, Vol. 154 (2017), Iss. P.322
https://doi.org/10.1016/j.compfluid.2017.03.006 [Citations: 26] -
Adjoint-based error estimation and mesh adaptation for the correction procedure via reconstruction method
Shi, Lei | Wang, Z.J.Journal of Computational Physics, Vol. 295 (2015), Iss. P.261
https://doi.org/10.1016/j.jcp.2015.04.011 [Citations: 28] -
On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations
Wang, Lai | Yu, Meilin2018 AIAA Aerospace Sciences Meeting, (2018),
https://doi.org/10.2514/6.2018-1095 [Citations: 3] -
Aeroacoustics noise prediction for the airfoil-rod benchmark using high-order large eddy simulation on unstructured grids and the acoustic analogy approach in frequency-domain
Modi, Rishit | Alhawwary, Mohammad A. | Akbarzadeh, Amir | Witherden, Freddie | Jameson, AntonyAIAA SCITECH 2023 Forum, (2023),
https://doi.org/10.2514/6.2023-0978 [Citations: 0] -
Landmarks and new frontiers of computational fluid dynamics
Shang, Joseph J. S.
Advances in Aerodynamics, Vol. 1 (2019), Iss. 1
https://doi.org/10.1186/s42774-019-0003-x [Citations: 11] -
Multi-dimensional Limiting Strategy for Higher-order CFD Methods - Progress and Issue (Invited)
Kim, Chongam | Park, Jin Seok22nd AIAA Computational Fluid Dynamics Conference, (2015),
https://doi.org/10.2514/6.2015-3199 [Citations: 1] -
A $$P_N P_M{-} CPR $$ P N P M - C P R Framework for Hyperbolic Conservation Laws
Shi, Lei | Wang, Z. J. | Zhang, L. P. | Liu, Wei | Fu, SongJournal of Scientific Computing, Vol. 61 (2014), Iss. 2 P.281
https://doi.org/10.1007/s10915-014-9829-x [Citations: 0] -
A priori and a posteriori evaluations of sub-grid scale models for the Burgers’ equation
Li, Yanan | Wang, Z.J.Computers & Fluids, Vol. 139 (2016), Iss. P.92
https://doi.org/10.1016/j.compfluid.2016.04.015 [Citations: 34] -
High-Order Methods for Computational Fluid Dynamics: A Brief Review of Compact Differential Formulation on Unstructured Grids
Huynh, HT | Wang, Zhi J. | Vincent, Peter21st AIAA Computational Fluid Dynamics Conference, (2013),
https://doi.org/10.2514/6.2013-2564 [Citations: 1] -
Adjoint-based Mesh Adaptation for the 3D Navier-Stokes Equations with the High-order CPR Method
Shi, Lei | Wang, Zhi J.54th AIAA Aerospace Sciences Meeting, (2016),
https://doi.org/10.2514/6.2016-1829 [Citations: 0] -
On the mesh resolution of industrial LES based on the DNS of flow over the T106C turbine
Alhawwary, Mohammad | Wang, Z.J.Advances in Aerodynamics, Vol. 1 (2019), Iss. 1
https://doi.org/10.1186/s42774-019-0023-6 [Citations: 14] -
DNS and LES of the flow over the T106C turbine using the high-order FR/CPR method
Alhawwary, Mohammad A. | Wang, Zhi J.AIAA Scitech 2020 Forum, (2020),
https://doi.org/10.2514/6.2020-1572 [Citations: 4] -
Development of a Stable High-Order Point-Value Enhanced Finite Volume (PFV) Method Based on Approximate Delta Functions
Xuan, Li-Jun | Majdalani, Joseph2018 Fluid Dynamics Conference, (2018),
https://doi.org/10.2514/6.2018-4168 [Citations: 0] -
Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation
Wang, Lai | Yu, MeilinJournal of Scientific Computing, Vol. 83 (2020), Iss. 2
https://doi.org/10.1007/s10915-020-01222-z [Citations: 12] -
A point-value enhanced finite volume method based on approximate delta functions
Xuan, Li-Jun | Majdalani, JosephJournal of Computational Physics, Vol. 355 (2018), Iss. P.37
https://doi.org/10.1016/j.jcp.2017.10.059 [Citations: 5] -
A Direct Discontinuous Galerkin Method for Computation of Turbulent Flows on Hybrid Grids
Cheng, Jian | Liu, Xiaodong | Yang, Xiaoquan | Liu, Tiegang | Luo, Hong46th AIAA Fluid Dynamics Conference, (2016),
https://doi.org/10.2514/6.2016-3333 [Citations: 3] -
A priori and a posteriori evaluations of subgrid stress models with the Burgers' equation
Li, Yanan | Wang, Zhi J.53rd AIAA Aerospace Sciences Meeting, (2015),
https://doi.org/10.2514/6.2015-1283 [Citations: 8] -
Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction
Park, Jin Seok | Chang, Tae Kyu | Kim, Chongam52nd Aerospace Sciences Meeting, (2014),
https://doi.org/10.2514/6.2014-0772 [Citations: 6] -
High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids
Huynh, H.T. | Wang, Z.J. | Vincent, P.E.Computers & Fluids, Vol. 98 (2014), Iss. P.209
https://doi.org/10.1016/j.compfluid.2013.12.007 [Citations: 173] -
A direct discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids
Cheng, Jian | Yang, Xiaoquan | Liu, Xiaodong | Liu, Tiegang | Luo, HongJournal of Computational Physics, Vol. 327 (2016), Iss. P.484
https://doi.org/10.1016/j.jcp.2016.09.049 [Citations: 39] -
Stable, non-dissipative, and conservative flux-reconstruction schemes in split forms
Abe, Yoshiaki | Morinaka, Issei | Haga, Takanori | Nonomura, Taku | Shibata, Hisaichi | Miyaji, KojiJournal of Computational Physics, Vol. 353 (2018), Iss. P.193
https://doi.org/10.1016/j.jcp.2017.10.007 [Citations: 33] -
Energy stable flux reconstruction schemes for advection–diffusion problems
Castonguay, P. | Williams, D.M. | Vincent, P.E. | Jameson, A.Computer Methods in Applied Mechanics and Engineering, Vol. 267 (2013), Iss. P.400
https://doi.org/10.1016/j.cma.2013.08.012 [Citations: 82] -
A review of flux reconstruction or correction procedure via reconstruction method for the Navier-Stokes equations
WANG, Z. J. | HUYNH, H. T.Mechanical Engineering Reviews, Vol. 3 (2016), Iss. 1 P.15-00475
https://doi.org/10.1299/mer.15-00475 [Citations: 45] -
High-order computational fluid dynamics tools for aircraft design
Wang, Z. J.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 372 (2014), Iss. 2022 P.20130318
https://doi.org/10.1098/rsta.2013.0318 [Citations: 34] -
Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction
Park, Jin Seok | Kim, ChongamJournal of Computational Physics, Vol. 308 (2016), Iss. P.57
https://doi.org/10.1016/j.jcp.2015.12.020 [Citations: 34] -
Estimation of Impacts of Removing Arbitrarily Constrained Domain Details to the Analysis of Incompressible Fluid Flows
Zhang, Kai | Li, Ming | Li, JingzhiCommunications in Computational Physics, Vol. 20 (2016), Iss. 4 P.944
https://doi.org/10.4208/cicp.071015.050216a [Citations: 3] -
Numerical Simulation of Transitional Flow Past a Wing Using Flux Reconstruction Method
Skarolek, Vilem | Miyaji, KojiAIAA Journal, Vol. 54 (2016), Iss. 4 P.1424
https://doi.org/10.2514/1.J054324 [Citations: 2] -
High-order point-value enhanced finite volume method for two-dimensional hyperbolic equations on unstructured meshes
Xuan, Li-Jun | Majdalani, JosephJournal of Computational Physics, Vol. 423 (2020), Iss. P.109756
https://doi.org/10.1016/j.jcp.2020.109756 [Citations: 2] -
Investigation of aspect ratio effects on flow characteristics and vorticity generation in shock-induced rectangular bubble
Singh, Satyvir
European Journal of Mechanics - B/Fluids, Vol. 101 (2023), Iss. P.131
https://doi.org/10.1016/j.euromechflu.2023.05.006 [Citations: 9] -
Quadrature-Free Implementation of a Discontinuous Galerkin Global Shallow-Water Model via Flux Correction Procedure
Nair, Ramachandran D.
Monthly Weather Review, Vol. 143 (2015), Iss. 4 P.1335
https://doi.org/10.1175/MWR-D-14-00174.1 [Citations: 3] -
Shock capturing with the high‐order flux reconstruction method on adaptive meshes based on p4est
Fu, Hao | Xia, Jian | Ma, XiuqiangEngineering Reports, Vol. 4 (2022), Iss. 10
https://doi.org/10.1002/eng2.12501 [Citations: 1] -
Adjoint-based Adaptation for the Correction Procedure via Reconstruction Method on Hybrid Meshes
Shi, Lei | Wang, Zhi J.53rd AIAA Aerospace Sciences Meeting, (2015),
https://doi.org/10.2514/6.2015-1740 [Citations: 1]