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Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods

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Abstract

The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K=kh→0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

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References

  1. Abboud, N.N., Pinsky, P.M.: Finite-element dispersion analysis for the 3-dimensional 2nd-order scalar wave-equation. Int. J. Numer. Methods Eng. 35, 1183–1218 (1992)

    Article  MATH  Google Scholar 

  2. Ainsworth, M.: Dispersive and dissipative behavior of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42, 553–575 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27, 5–60 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ainsworth, M., Wajid, H.A.: Dispersive and dissipative behavior of the spectral element method. SIAM J. Numer. Anal. 47, 3910–3937 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ainsworth, M., Wajid, H.A.: Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes. In: Computer Methods in Mechanics, vol. 1, pp. 3–17 (2010)

    Chapter  Google Scholar 

  7. Chavent, G., Salzano, G.: A finite element method for the 1d water flooding problem with gravity. J. Comput. Phys. 45, 307–344 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Shu, C.W.: The Runge-Kutta local projection P 1-discontinuous Galerkin method for scalar conservation laws. Modél. Math. Anal. Numér. 25, 337–361 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Hesthaven, J.S., Warburton, T.: Nodal high order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, F., Atkins, H.: Eigensolution analysis of the discontinuous Galerkin method with non-uniform grids, part I: one space dimension. J. Comput. Phys. 182, 516–545 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, F., Hussaini, M., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151, 921–946 (1999)

    Article  MATH  Google Scholar 

  13. Ihlenburg, F., Babuška, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38, 3745–3774 (1995)

    Article  MATH  Google Scholar 

  14. Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)

    Article  MATH  Google Scholar 

  15. Lesaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–123. Academic Press, New York (1974)

    Google Scholar 

  16. Peterson, T.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28, 133–140 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Qiu, J.: A numerical comparison of the Lax-Wendroff discontinuous Galerkin method based on different numerical fluxes. J. Sci. Comput. 30, 345–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qiu, J., Michael, D., Shu, C.-W.: The discontinuous Galerkin method with Lax-Wendroff type time discretizations. Comput. Methods Appl. Mech. Eng. 194, 4528–4543 (2005)

    Article  MATH  Google Scholar 

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

  20. Richter, G.R.: An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50, 75–88 (1988)

    Article  MATH  Google Scholar 

  21. Sármány, D., Botchev, M.A., van der Vegt, J.J.W.: Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretizations of the Maxwell equations. J. Sci. Comput. 33, 47–74 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sherwin, S.: Dispersive Analysis of the Continuous and Discontinuous Galerkin Formulations. Lecture notes

  23. Stanescu, D., Kopriva, D.A., Hussaini, M.Y.: Dispersive analysis for discontinuous spectral element methods. J. Sci. Comput. 15, 149–171 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, Q., Shu, C.-W.: Error estimates to smooth solution of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)

    Article  MathSciNet  Google Scholar 

  25. Zhang, Q., Shu, C.-W.: Error estimates to smooth solution of Runge-Kutta discontinuous Galerkin methods for symmetrizable conservation laws. SIAM J. Numer. Anal. 44, 1702–1720 (2006)

    Article  MathSciNet  Google Scholar 

  26. Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimates to the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48(2), 772–795 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhong, X., Shu, C.-W.: Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200, 2814–2827 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180, USA

    He Yang & Fengyan Li

  2. School of Mathematical Science, Xiamen University, Xiamen, Fujian, 361005, P.R. China

    Jianxian Qiu

Authors
  1. He Yang
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  2. Fengyan Li
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  3. Jianxian Qiu
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Correspondence to Fengyan Li.

Additional information

The research of F. Li and H. Yang is partially supported by NSF CAREER award DMS-0847241 and an Alfred P. Sloan Research Fellowship. The research of J. Qiu is partially supported by the National Science Foundation of China Grant No. 10931004 and ISTCP of China Grant No. 2010DFR00700.

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Yang, H., Li, F. & Qiu, J. Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods. J Sci Comput 55, 552–574 (2013). https://doi.org/10.1007/s10915-012-9647-y

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