Skip to main content
Springer Nature Link
Log in

Time stepping in discontinuous Galerkin method

  • Published:
Journal of Hydrodynamics Aims and scope Submit manuscript

Abstract

The time discretization in the Discontinuous Galerkin (DG) scheme has been traditionally based on the Total Variation Diminishing (TVD) second-order Runge-Kutta (RK2) scheme. Computational efficiency and accuracy with the Euler Forward (EF) and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are conducted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.

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

Access this article

Log in via an institution

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

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

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

References

  1. CHEN Er-yun, MA Da-wei and LE Gui-gao et al. Numerical simulation of highly underexpanded axisymmetric jet with Runge-Kutta discontinuous Galerkin finite element method[J]. Journal of Hydrodynamics, 2008, 20(5): 617–623.

    Article  Google Scholar 

  2. LAI Wencong, KHAN Abdul A. Modeling dam-break flood over natural rivers using discontinuous Galerkin method[J]. Journal of Hydrodynamics, 2012, 24(4): 467–478.

    Article  Google Scholar 

  3. QIU J., DUMBSER M. and SHU C. W. The discontinuous Galerkin method with Lax-Wendroff type time discretizations[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(42-47): 4528–4543.

    Article  MathSciNet  Google Scholar 

  4. GASSNER G., DUMBSER M. and HINDENLANG F. et al. Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors[J]. Journal of Computational Physics, 2011, 230(11): 4232–4247.

    Article  MathSciNet  Google Scholar 

  5. TRAHAN C. J., DAWSON C. Local time-stepping in Runge-Kutta discontinuous finite element methods applied to shallow-water equations[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 217-220: 139–152.

    MATH  Google Scholar 

  6. DAWSON C., TRAHAN C. J. and KUBATKO E. J. et al. A parallel local timestepping Runge-Kutta disconti nuous Galerkin method with applications to coastal ocean modeling[J]. Computer Methods in Applied Mechanics and Engineering, 2013,259(1): 154–165.

    Article  MathSciNet  Google Scholar 

  7. ZHONG X., SHU C. W. A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods[J]. Journal of Computational Physics, 2013, 232(1): 397–415.

    Article  MathSciNet  Google Scholar 

  8. ZHAO J., TANG H. Runge-Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics[J]. Journal of Computational Physics, 2013, 242(1): 138–168.

    Article  MathSciNet  Google Scholar 

  9. LI B. Q. Discontinuous finite elements in fluid dynamics and heat transfer[M]. London, UK: Springer-Verlag, 2006.

    Google Scholar 

  10. LAI W., KHAN A. A. A discontinuous Galerkin method for two-dimensional shallow water flows[J]. International Journal for Numerical Methods in Fluids, 2012, 70(8): 939–960.

    Article  MathSciNet  Google Scholar 

  11. LAI W., KHAN A. A. Discontinuous Galerkin method for 1D shallow water flow in nonrectangular and nonprismatic channels[J]. Journal of Hydraulic Engineering, ASCE, 2012, 138(3): 285–296.

    Article  Google Scholar 

  12. LAI W., KHAN A. A. Discontinuous Galerkin method for 1D shallow water surface flow with water surface slope limiter[J]. International Journal of Civil and Environmental Engineering, 2011, 3(3): 167–176.

    Google Scholar 

  13. LAI W., KHAN A. A. Discontinuous Galerkin method for 1D shallow water flows in natural rivers[J]. Engineering Applications of Computational Fluid Mechanics, 2012, 6(1): 74–86.

    Article  Google Scholar 

  14. LIN G., LAI J. and GUO W. High-resolution TVD schemes in finite volume method for hydraulic shock wave modeling[J]. Journal of Hydraulic Research, 2005, 43(4): 376–389.

    Article  Google Scholar 

  15. SOARES FRAZÃO S., ZECH Y. Dam-break in channel with 900 bend[J]. Journal of Hydraulic Engineering, ASCE, 2002, 128(11): 956–968.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Glenn Department of Civil Engineering, Clemson University, Clemson, S.C., USA

    Wencong Lai & Abdul A. Khan

Authors
  1. Wencong Lai
    View author publications

    Search author on:PubMed Google Scholar

  2. Abdul A. Khan
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Wencong Lai.

Additional information

Biography: LAI Wencong (1985-), Male, Ph. D.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lai, W., Khan, A.A. Time stepping in discontinuous Galerkin method. J Hydrodyn 25, 321–329 (2013). https://doi.org/10.1016/S1001-6058(11)60370-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/S1001-6058(11)60370-4

Key words