Skip to main content
SpringerLink
Log in

A \(P_N P_M{-} CPR \) Framework for Hyperbolic Conservation Laws

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The correction procedure via reconstruction (CPR) method is a discontinuous nodal formulation unifying several well-known methods in a simple finite difference like manner. The \(P_NP_M{-} CPR \) formulation is an extension of \(P_NP_M\) or the reconstructed discontinuous Galerkin (RDG) method to the CPR framework. It is a hybrid finite volume and discontinuous Galerkin (DG) method, in which neighboring cells are used to build a higher order polynomial than the solution representation in the cell under consideration. In this paper, we present several \(P_NP_M\) schemes under the CPR framework. Many interesting schemes with various orders of accuracy and efficiency are developed. The dispersion and dissipation properties of those methods are investigated through a Fourier analysis, which shows that the \(P_NP_M{-} CPR \) method is dependent on the position of the solution points. Optimal solution points for 1D \(P_NP_M{-} CPR \) schemes which can produce expected order of accuracy are identified. In addition, the \(P_NP_M{-} CPR \) method is extended to solve 2D inviscid flow governed by the Euler equations and several numerical tests are performed to assess its performance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, vol. 54. Springer, New York (2008)

    Google Scholar 

  2. Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for CFD. Oxford University Press, USA (1999)

    MATH  Google Scholar 

  3. Wang, Z.J.: Adaptive High-Order Methods in Computational Fluid Dynamics, vol. 2. World Scientific, Singapore (2011)

    MATH  Google Scholar 

  4. Ekaterinaris, J.: High-order accurate, low numerical diffusion methods for aerodynamics. Prog. Aerosp. Sci. 41, 192–300 (2005)

    Article  Google Scholar 

  5. Wang, Z.J.: High-order methods for the Euler and Navier–Stokes equations on unstructured grids. Prog. Aerosp. Sci. 43, 1–41 (2007)

    Article  Google Scholar 

  6. Barth, T., Frederickson, P.: Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper 90, 0013 (1990)

    Google Scholar 

  7. Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2d Euler equations. J. Comput. Phys. 138, 251–285 (1997a)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method forthe numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997b)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bassi, F., Rebay, S.: Gmres discontinuous galerkin solution of the compressible Navier–Stokes equations. Lect. Notes Comput. Sci. Eng. 11, 197–208 (2000)

    Article  MathSciNet  Google Scholar 

  10. Cockburn, B., Lin, S., Shu, C.-W.: Tvb runge-kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998a)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Shu, C.-W.: The local discontinuous galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998b)

    Google Scholar 

  13. Raalte, M.V., Leer, B.V.: Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Commun. Comput. Phys. 5, 683–693 (2009)

    MathSciNet  Google Scholar 

  14. Gassner, G., Lörcher, F., Munz, C., Hesthaven, J.: Polymorphic nodal elements and their application in discontinuous Galerkin methods. J. Comput. Phys. 228, 1573–1590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peraire, J., Persson, P.: The compact discontinuous galerkin (cdg) method for elliptic problems. Arxiv, Preprint arXiv:math/0702353 (2007)

  16. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479 (1973)

  17. Leer, B.V.: Towards the ultimate conservative difference scheme V. A second order sequel to godunovs method. J. Comput. Phys. 32, 101–136 (1979)

    Article  Google Scholar 

  18. Warburton, T.: An explicit construction of interpolation nodes on the simplex. J. Eng. Math. 56, 247–262 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, Y., Vinokur, M., Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys. 212, 454–472 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178, 210–251 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, Z., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys. 179, 665–697 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kopriva, D., Kolias, J.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244–261 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. May, G., Jameson, A.: A spectral difference method for the euler and Navier–Stokes equations on unstructured meshes. AIAA Paper 304, 2006 (2006)

    Google Scholar 

  24. Van den Abeele, K., Lacor, C., Wang, Z.: On the stability and accuracy of the spectral difference method. J. Sci. Comput. 37, 162–188 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liang, C., Jameson, A., Wang, Z.J.: Spectral difference method for compressible flow on unstructured grids with mixed elements. J. Comput. Phys. 228, 2847–2858 (2009)

    Article  MATH  Google Scholar 

  26. Haga, T., Gao, H., Wang, Z.J.: A high-order unifying discontinuous formulation for the Navier–Stokes equations on 3d mixed grids. Math. Model. Nat. Phenom. 6, 28–56 (2011)

    Google Scholar 

  27. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper, 4079, 2007 (2007)

  28. Huynh, H.T.: A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. AIAA Paper, 403, 2009 (2009)

  29. Huynh, H.T.: High-order methods by correction procedures using reconstructions. Adapt. High-Order Methods Comput. Fluid Dyn. 2, 391–422 (2011)

    Article  MathSciNet  Google Scholar 

  30. Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228, 8161–8186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, Z.J., Gao, H., Haga, T.: A unifying discontinuous formulation for hybrid meshes. Adapt. High-Order Methods Comput. Fluid Dyn. 2, 423–453 (2011)

    Google Scholar 

  32. Cagnone, J., Vermeire, B., Nadarajah, S.: A p-adaptive LCP formulation for the compressible Navier–Stokes equations. J. Comput. Phys. 233, 324–338 (2013)

    Article  MathSciNet  Google Scholar 

  33. Dumbser, M., Balsara, D., Toro, E., Munz, C.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227, 8209–8253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dumbser, M.: PNPM schemes on unstructured meshes for time-dependent partial differential equations. Adapt. High-Order Methods Comput. Fluid Dyn. 2, 203 (2011)

    Article  MathSciNet  Google Scholar 

  35. Luo, H., Luo, L., Nourgaliev, R., Mousseau, V., Dinh, N.: A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids. J. Comput. Phys. 229, 6961–6978 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Luo, H., Luo, L., Nourgaliev, R.: A reconstructed discontinuous galerkin method for the Euler equations on arbitrary grids. Commun. Comput. Phys. 12, 1495–1519 (2012)

    MathSciNet  Google Scholar 

  37. Zhang, L., Liu, W., He, L., Deng, X., Zhang, H.: A class of hybrid dg/fv methods for conservation laws I: Basic formulation and one-dimensional systems. J. Comput. Phys. 231, 1081–1103 (2012a)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, L., Liu, W., He, L., Deng, X., Zhang, H.: A class of hybrid dg/fv methods for conservation laws II: two-dimensional cases. J. Comput. Phys. 231, 1104–1120 (2012b)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, L., Liu, W., He, L., Deng, X.: A class of hybrid dg/fv methods for conservation laws III: two-dimensional euler equations. Commun. Comput. Phys. 12, 284–314 (2012c)

    MathSciNet  Google Scholar 

  40. Xuan, L.-J., Wu, J.-Z.: A weighted-integral based scheme. J. Comput. Phys. 229, 5999–6014 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Haga, T., Gao, H., Wang, Z.: A high-order unifying discontinuous formulation for the Navier–Stokes equations on 3d mixed grids. Math. Model. Nat. Phenom. 6, 28–56 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gao, H., Wang, Z.J.: A conservative correction procedure via reconstruction formulation with the chain-rule divergence evaluation. J. Comput. Phys. 232, 7–13 (2013a)

    Article  MathSciNet  Google Scholar 

  43. Gao, H., Wang, Z.J.: Differential formulation of discontinuous Galerkin and related methods for the Navier–Stokes equation. Commun. Comput. Phys. 13, 1013–1044 (2013b)

    MathSciNet  Google Scholar 

  44. Castonguay, P., Vincent, P.E., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes for triangular elements. J. Sci. Comput. 51, 224–256 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Jameson, A., Vincent, P.E., Castonguay, P.: On the non-linear stability of flux reconstruction schemes. J. Sci. Comput. 50, 434–445 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MATH  Google Scholar 

  47. Taylor, M., Wingate, B.A: Several new quadrature formulas for polynomial integration in the triangle. ArXiv Mathematics e-prints (2005)

  48. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Technical Report (1997)

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Kansas, Lawrence, KS , 66045, USA

    Lei Shi & Z. J. Wang

  2. State Key Laboratory of Aerodynamics, Mianyang , 621000, Sichuan, China

    L. P. Zhang & Wei Liu

  3. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Song Fu

Authors
  1. Lei Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. J. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. P. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Song Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Shi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shi, L., Wang, Z.J., Zhang, L.P. et al. A \(P_N P_M{-} CPR \) Framework for Hyperbolic Conservation Laws. J Sci Comput 61, 281–307 (2014). https://doi.org/10.1007/s10915-014-9829-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9829-x

Keywords

Search

Navigation