Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T08:59:27.788Z Has data issue: false hasContentIssue false
  • English
  • Français

A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations

Published online by Cambridge University Press:  20 August 2015

Laiping Zhang ,
Wei Liu ,
Lixin He  and
Xiaogang Deng
Laiping Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Wei Liu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Lixin He*
Affiliation:
Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Xiaogang Deng*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
*
Corresponding author.Email:zhanglp_cardc@126.com
Email:lw4992@gmail.com
Email:hlx_cardc@126.com
Email:xgdeng@skla.cardc.cn
Get access

Abstract

A concept of “static reconstruction” and “dynamic reconstruction” was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a “hybrid reconstruction” approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piece-wise polynomial are computed locally in a cell by the traditional DG method (called as “dynamic reconstruction”), while the higher-order derivatives are re-constructed by the “static reconstruction” of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.

Keywords

76G2576H0576M1076M1276M25Discontinuous Galerkin methodfinite volume methodreconstructionhybrid method
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 284 - 314
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high-order essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), 231–303.Google Scholar
[2]Jiang, G. and Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[3]Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[4]Cockburn, B. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52 (1989), 411–435.Google Scholar
[5]Cockburn, B., Lin, S. Y. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[6]Cockburn, B., Hou, S. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54 (1990), 545–581.Google Scholar
[7]Cockburn, B. and Shu, C. W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., (2001), 2001–173.Google Scholar
[8]Abgrall, A., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114 (1994), 45–58.CrossRefGoogle Scholar
[9]Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144 (1998), 194–212.Google Scholar
[10]Hu, C. and Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), 97–127.Google Scholar
[11]Dumbser, M. and Kaser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221 (2007), 693–723.Google Scholar
[12]Dumbser, M., Kaser, M., Titarev, V. and Toro, E., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226 (2007), 204–243.Google Scholar
[13]Zhang, Y. and Shu, C. W., Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys., 5 (2009), 836–848.Google Scholar
[14]Titarev, V. A., Tsoutsanis, P. and Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8(3) (2010), 2010–585.Google Scholar
[15]Tsoutsanis, P., Titarev, V. and Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230 (2011), 1585–1601.Google Scholar
[16]Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210–251.Google Scholar
[17]Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation lawson unstructured grids ii: extension to two-dimensional scalar equation, J. Comput. Phys., 179 (2002), 665–697.Google Scholar
[18]Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation lawson unstructured grids III: one-dimensional systems and partition optimization, J. Sci. Comput., 20 (2004), 137–157.Google Scholar
[19]Wang, Z. J., Zhang, L. P. and Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations, J. Comput. Phys., 194 (2004), 716–741.Google Scholar
[20]Liu, Y., Vinokur, M. and Wang, Z. J., Discontinuous spectral difference method for conservation laws on unstructured grids, J. Comput. Phys., 216 (2006), 780–801.CrossRefGoogle Scholar
[21]Venkatakrishnan, V., Allmaras, S. R., Kamenetskii, D. S. and Johnson, F. T., Higher order schemes for the compressible Navier-Stokes equations, AIAA-2003-3987, 2003.CrossRefGoogle Scholar
[22]May, G. and Jameson, A., A spectral difference method for the Euler and Navier-Stokes equations, AIAA-2006-304, 2006.CrossRefGoogle Scholar
[23]Ekaterinaris, J. A., High-order accurate, low numerical diffusion methods for aerodynamics, Prog. Aero. Sci., 41 (2005), 192–300.Google Scholar
[24]Wang, Z. J., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Prog. Aero. Sci., 43 (2007), 1–41.Google Scholar
[25]Baker, T. J., Mesh generation: art or science?, Prog. Aero. Sci., 41 (2005), 29–63.CrossRefGoogle Scholar
[26]Cockburn, B., Karniadakis, G. E. and Shu, C. W., Discontinuous Galerkin Methods, Berlin, Springer, 2000.Google Scholar
[27]Luo, H., Baum, J. D. and Lohner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227 (2008), 8875–8893.Google Scholar
[28]Barth, T. and Frederickson, P., High order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper 90-0013, 1990.Google Scholar
[29]Thareja, R. R. and Stewart, J. R., A point implicit unstructured grid solver for the Euler and Navier-Stokes equations, Int. J. Num. Meth. Fluids, 9 (1989), 405–425.Google Scholar
[30]Luo, H., Baum, J. D. and Lohner, R., High-Reynolds number viscous computations using an unstructured-grid method, J. Aircraft, 42 (2005), 483–492.Google Scholar
[31]He, L. X., Zhang, L. P. and Zhang, H. X., A finite element/finite volume mixed solver on hybrid grids, Proceedings of the Fourth International Conference on Computational Fluid Dynamics, 10-14 July, 2006, Ghent, Belgium, edited by Deconinck, Herman and Dick, Erik, Springer Press, (2006), 2006–695.Google Scholar
[32]Dumbser, M., Balsara, D. S. and Toro, E. F., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227 (2008), 8209–8253.Google Scholar
[33]Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids, 39 (2010), 60–76.Google Scholar
[34]Dumbser, M. and Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J. Comput. Phys., 228 (2009), 6991–7006.Google Scholar
[35]Qiu, J. X. and Shu, C. W., Hermite WENO schemes and their application aslimiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case, J. Comput. Phys., 193 (2003), 115–135.Google Scholar
[36]Qiu, J. and Shu, C. W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two-dimensional case, Comput. Fluids, 34 (2005), 642–663.Google Scholar
[37]Luo, H., Luo, L. P., Nourgaliev, R., Mousseau, V. A. and Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys., 229 (2010), 6961–6978.Google Scholar
[38]Luo, H., Luo, L. P., Ali, A., Norgaliev, R. and Cai, C., A parallel, reconstructed discontinuous Galerkin method for the compressible flows on arbitrary grids, Commun. Comput. Phys., 9(2) (2011), 2011–363.Google Scholar
[39]Zhang, L. P., Liu, W., He, L. X. and Deng, X. G., A new class of DG/FV hybrid schemes for one-dimensional conservation law, the 8th Asian Conference on Computational Fluid Dynamics, Hong Kong, 10-14 January, 2010.Google Scholar
[40]Zhang, L. P., Liu, W., He, L. X., Deng, X. G. and Zhang, H. X., A class of hybrid DG/FV methods for conservation laws I: basic formulation and one-dimensional systems, J. Comput. Phys., 2011, in press.Google Scholar
[41]Zhang, L. P., Liu, W., He, L. X., Deng, X. G. and Zhang, H. X., A1 class of hybrid DG/FV method for conservation laws II: two-dimensional cases, J. Comput. Phys., 2011, in press.Google Scholar
[42]Huynh, H. T., A flux reconstruction approachto high-order schemes including discontinuous Galerkin methods, AIAA-2007-4079,2007.Google Scholar
[43]Wang, Z. J. and Gao, H., A unifying lifting collocation penalty formulation for the Euler equations on mixed grids, AIAA-2009-0401, 2009.Google Scholar
[44]Liu, Y. J., Shu, C. W., Tadmor, E. and Zhang, M. P., Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal., 45 (2007), 2442–2467.Google Scholar
[45]Luo, H., Baum, J. D. and Lhner, R., A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids, J. Comput. Phys., 211 (2006), 767–783.Google Scholar
[46]Nastase, C. R. and Mavriplis, D. J., High-order discontinuous Galerkin methods using an hp-multigrid approach, J. Comput. Phys., 213 (2006), 330–357.Google Scholar
[47]Yang, M. and Wang, Z. J., A parameter-free generalized moment limiter for high-order methods on unstructured grids, AIAA-2009-605, 2009.Google Scholar
[48]Krivodonova, L. and Berger, M., High-order accurate implementation of solid wall boundary condition in curved geometries, J. Comput. Phys., 211 (2006), 492–512.Google Scholar
[49]Barth, T. J. and Jesperson, D. C., The design of application of upwind schemes on unstructured grids, AIAA-1989-0366, 1989.Google Scholar
[50]Luo, H., Baum, J. D. and Lohner, R., A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids, J. Comput. Phys., 225 (2007), 686–713.Google Scholar