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A New Lax–Wendroff Discontinuous Galerkin Method with Superconvergence

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Abstract

Superconvergence of discontinuous Galerkin (DG) methods for hyperbolic conservation laws has been intensively studied in different settings in the past. For example, the numerical solution by a semi-discrete DG scheme is superconvergent with order \(2k+1\) in the negative-order norms, when the solution is globally smooth. Hence the accuracy of the numerical solution can be enhanced to \((2k+1)\)th order accuracy by simply applying a carefully designed post-processor (Cockburn et al. in Math Comput 72:577–606, 2003). In this paper, we investigate superconvergence for the DG schemes coupled with Lax–Wendroff (LW) time discretization (LWDG). Through numerical experiments, we find that the original LWDG scheme developed in Qiu et al. (Comput Methods Appl Mech Eng 194:4528–4543, 2005) does not exhibit superconvergence properties mentioned above. In order to restore superconvergence, we propose to use the techniques from the local DG scheme to reconstruct high order spatial derivatives, while, in the original LWDG formulation, the high order derivatives are obtained by direct differentiation of the numerical solution. A collection of 1-D and 2-D numerical experiments are presented to verify superconvergence properties of the newly proposed LWDG scheme. We also perform Fourier analysis via symbolic computations to investigate the superconvergence of the proposed scheme.

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References

  1. Adjerid, S., Klauser, A.: Superconvergence of discontinuous finite element solutions for transient convection–diffusion problems. J. Sci. Comput. 22, 5–24 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. 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, 5–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bramble, J., Schatz, A.: Higher order local accuracy by averaging in the finite element method. Math. Comput. 31, 94–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, W. , Zhang, Z., and Zou, Q.: Superconvergence of discontinuous Galerkin method for linear hyperbolic equations. arXiv:1311.6938

  6. Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227, 9612–9627 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, Y., Shu, C.-W.: Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations. Comput. Struct. 87, 630–641 (2009)

    Article  Google Scholar 

  8. Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection–diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Luskin, M., Shu, C.-W., Suli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72, 577–606 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Cockburn, B., Shu, C.-W.: The Runge–Kutta local projection \(P^{1}\)-discontinuous Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Numer. Anal. 25, 337–361 (1991)

    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 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, W., Li, F., Qiu, J.-X.: Local-structure-preserving discontinuous Galerkin methods with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations. J. Sci. Comput. 47, 239–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, W., Zhong, X., Qiu, J.-M.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235, 458–485 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Ji, L., Xu, Y., Ryan, J.: Accuracy-enhancement of discontinuous Galerkin solutions for convection–diffusion equations in multiple-dimensions. Math. Comput. 81, 1929–1950 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ji, L., Xu, Y., Ryan, J.: Negative-order norm estimates for nonlinear hyperbolic conservation laws. J. Sci. Comput. 54, 531–548 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, G., Shu, C.-W.: On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62, 531–538 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Qiu, J.-X., Shu, C.-W.: Finite difference WENO schemes with Lax–Wendroff type time discretizations. SIAM J. Sci. Comput. 24, 2185–2198 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Qiu, J.-X., Shu, C.-W.: Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26, 907–929 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Sherwin, S.: Dispersion analysis of the continuous and discontinuous Galerkin formulation. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 425–432 (2000)

  27. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)

    MathSciNet  Google Scholar 

  28. Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, H., Li, F., Qiu, J.-X.: Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput. 55, 552–574 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 Mathematics, University of Houston, Houston, TX, 77004, USA

    Wei Guo & Jing-Mei Qiu

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen, 361005, Fujian, China

    Jianxian Qiu

Authors
  1. Wei Guo
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  2. Jing-Mei Qiu
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  3. Jianxian Qiu
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Correspondence to Jianxian Qiu.

Additional information

The research was partially supported by NSFC Grant 91230110 and 11328104.

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Guo, W., Qiu, JM. & Qiu, J. A New Lax–Wendroff Discontinuous Galerkin Method with Superconvergence. J Sci Comput 65, 299–326 (2015). https://doi.org/10.1007/s10915-014-9968-0

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  • DOI: https://doi.org/10.1007/s10915-014-9968-0

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