Skip to main content
Log in

Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Cockburn, B., Karniadakis, G., Shu, C.W.: The development of discontinuous Galerkin method. In: Lecture Notes in Computational Science and Engineering. Cockburn, B., Karniadakis, G., Shu, C.W. eds. Discontinuous Galerkin Methods, Theory, Computation, and Applications, Springer, New York, 11, 5–50 (2000)

    Google Scholar 

  3. Hu, C.Q., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wang, R.Q., Shen, Y.Q.: Some weight-type high-resolution difference schemes and their applications. Acta Mech. Sin. 15, 313–324 (1999)

    Article  MathSciNet  Google Scholar 

  5. Qiu, J., Shu, C.W.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One dimensional case. J. Comput. Phys. 193, 115–135 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Qiu, J., Shu, C.W.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case. Computers and Fluids 34, 642–663 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Luo, H., Baum, J.D., Lohner, R.: A HermiteWENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225, 686–713 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhu, J., Qiu, J.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method III: Unstructured meshes. J. Sci. Comput. 39, 293–321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jiang, Z., Yan, C., Yu, J. et al.: Hermite WENO-based limiters for high order discontinuous Galerkin method on unstructured grids. Acta Mechanica Sinica. 28, 1–12 (2012)

    Article  MathSciNet  Google Scholar 

  10. Zhu, J., Qiu, J.: Local DG method using WENO type limiters for convection-diffusion problems. J. Comput. Phys. 230, 4353–4375 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bassi, F., Rebay, S.: GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations. In: Lecture Note in Computational Science and Engineering. Cockburn, B., Karniadakis, G., Shu, C.W. eds. Discontinuous Galerkin Methods, Theory, Computation, and Applications, Springer-Verlag, New York, 11, 197–208 (2000)

    Google Scholar 

  12. Fidkowski, K.J., Oliver, T.A., Lu, J. et al.: P-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 207, 92–113 (2005)

    Article  MATH  Google Scholar 

  13. Luo, H., Baum, J.D., Löhner, R.: A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys. 2211, 767–783 (2006)

    Article  Google Scholar 

  14. Nastase, C.R., Mavriplis, D.J.: High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys. 213, 330–357 (2006)

    Article  MATH  Google Scholar 

  15. Luo, H., Sharov, D., Baum, J.D. et al.: A class of matrix-free implicit methods for compressible flows on unstructured grids. In: Proceedings of the First International Conference on Computational Fluid Dynamics. Kyoto, Japan, 10–14 (2000)

    Google Scholar 

  16. Luo, H., Baum, J.D., Lohner, R.: A fast, matrix-free implicit method for compressible flows on unstructured grids. J. Comput. Phys. 146, 664–690 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, R.F., Wang, Z.J.: Fast block lower-upper symmetric Gauss Seidel scheme for arbitrary grids. AIAA J. 38, 2238–2245 (2000)

    Article  Google Scholar 

  18. Jameson, A., Caughey, D.A.: How many steps are required to solve the Euler equations of steady compressible flow. In: search of a fast solution algorithm. In: Proc. of 15th AIAA Computational Fluid Dynamics Conference. Anaheim, CA (2001)

    Google Scholar 

  19. Sun, Y., Wang, Z.J., Liu, Y.: Efficient implicit non-linear LUSGS approach for compressible flow computation using high-order spectral difference method. Communications in Computational Physics 5, 760–778 (2009)

    MathSciNet  Google Scholar 

  20. Haga, T., Sawada, K., Wang, Z.J.: An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids. Communications in Computational Physics 6, 978–996 (2009)

    Article  MathSciNet  Google Scholar 

  21. Parsani, M., Van den Abeele, K., Lacor, C. et al.: Implicit LUSGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier-Stokes equations. J. Comput. Phys. 229, 828–850 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Haga, T., Gao, H., Wang, Z.J.: Efficient solution techniques for high-order methods on 3D anisotropic hybrid meshes. AIAA paper 2011-45 (2011)

    Google Scholar 

  23. Bassi, F., Crivellini, A., Rebay, S., et al.: Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations. Computers and Fluids 34, 507–540 (2005)

    Article  MATH  Google Scholar 

  24. Johan, Z., Hughes, T.J.R., Shakib, F.: A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Computer Methods in Applied Mechanics and Engineering 87, 281–304 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nielsen, E.J., Anderson, W.K., Walters, R.W., et al.: Application of Newton-Krylov methodology to a three-dimensional unstructured Euler codes. AIAA paper 95-1733 (1995)

    Google Scholar 

  26. Yu, J., Yan, C.: Study on discontinuous Galerkin method for Navier-Stokes equations. Chinese Journal of Theoretical and Applied Mechanics 42, 664–690 (2010)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Wieting, A.R.: Experimental study of shock wave interference heating on a cylindrical leading edge. NASA TM 100484 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhen-Hua Jiang.

Additional information

The project was supported by the National Basic Research Program of China (2009CB724104).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, ZH., Yan, C. & Yu, J. Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations. Acta Mech Sin 29, 526–533 (2013). https://doi.org/10.1007/s10409-013-0042-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-013-0042-1

Keywords

Navigation