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Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

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Abstract

We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

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References

  1. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)

  2. 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  Google Scholar 

  3. Appelö, D., Hagstrom, T.: A new discontinuous Galerkin formulation for wave equations in second order form. SIAM J. Numer. Anal. 53(6), 2705–2726 (2015)

    Article  MathSciNet  Google Scholar 

  4. Appelö, D., Hagstrom, T.: An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form. Comput. Methods Appl. Mech. Eng. 338, 362–391 (2018)

    Article  MathSciNet  Google Scholar 

  5. Appelö, D., Hagstrom, T., Wang, Q., Zhang, L.: An energy-based discontinuous Galerkin method for semilinear wave equations. J. Comput. Phys. 418, 109608 (2020)

    Article  MathSciNet  Google Scholar 

  6. Banks, J.W., Buckner, B.B., Hagstrom, T., Juhnke, K.: Discontinuous Galerkin Galerkin differences for the wave equation in second-order form. SIAM J. Sci. Comput. 43(2), A1497–A1526 (2021)

  7. Banks, J.W., Hagstrom, T.: On Galerkin difference methods. J. Comput. Phys. 313, 310–327 (2016)

    Article  MathSciNet  Google Scholar 

  8. Banks, J.W., Henshaw, W.D.: Upwind schemes for the wave equation in second-order form. J. Comput. Phys. 231(17), 5854–5889 (2012). https://doi.org/10.1016/j.jcp.2012.05.012. http://www.sciencedirect.com/science/article/pii/S0021999112002367

  9. Chan, J., Hewett, R.J., Warburton, T.: Weight-adjusted discontinuous Galerkin methods: curvilinear meshes. SIAM J. Sci. Comput. 39(6), A2395–A2421 (2017)

    Article  MathSciNet  Google Scholar 

  10. Chan, J., Hewett, R.J., Warburton, T.: Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media. SIAM J. Sci. Comput. 39(6), A2935–A2961 (2017)

    Article  MathSciNet  Google Scholar 

  11. Chou, C.S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014). https://doi.org/10.1016/j.jcp.2014.04.009. http://www.sciencedirect.com/science/article/pii/S0021999114002721

  12. Grote, M.J., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2431 (2006). http://www.jstor.org/stable/40232901

  13. Hagstrom, T., Banks, J.W., Buckner, B.B., Juhnke, K.: Discontinuous Galerkin difference methods for symmetric hyperbolic systems. J. Sci. Comput. 81, 1509–1526 (2019)

    Article  MathSciNet  Google Scholar 

  14. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media, Berlin (2007)

    MATH  Google Scholar 

  15. Hu, F.Q., Hussaini, M., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151(2), 921–946 (1999)

    Article  Google Scholar 

  16. Jones, M., Plassmann, P.: Algorithm 740: Fortran subroutines to compute improved incomplete Cholesky factorizations. ACM Trans. Math. Soft. (TOMS) 21, 5–17 (1995)

    Article  Google Scholar 

  17. Kozdon, J., Wilcox, L., Hagstrom, T., Banks, J.: Robust approaches to handling complex geometries with Galerkin difference methods. J. Comput. Phys. 392, 483–510 (2019)

    Article  MathSciNet  Google Scholar 

  18. Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numer. Math. 6(1), 185–199 (1964)

    Article  MathSciNet  Google Scholar 

  19. Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficient. J. Sci. Comput. 51, 650–682 (2012)

    Article  MathSciNet  Google Scholar 

  20. Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)

    Article  MathSciNet  Google Scholar 

  21. Moura, R.C., Sherwin, S., Peiró, J.: Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods. J. Comput. Phys. 298, 695–710 (2015)

    Article  MathSciNet  Google Scholar 

  22. Riviere, B., Wheeler, M.: Discontinuous finite element methods for acoustic and elastic wave problems. Part i: semidiscrete error estimates. Contemp. Math. 329, 271–282 (2003)

    Article  Google Scholar 

  23. Schmitz, P.G., Ying, L.: A fast nested dissection solver for Cartesian 3D elliptic problems using hierarchical matrices. J. Comput. Phys. 258, 227–245 (2014)

    Article  MathSciNet  Google Scholar 

  24. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

    Article  MathSciNet  Google Scholar 

  25. Zhang, L., Hagstrom, T., Appelö, D.: An energy-based discontinuous Galerkin method for the wave equation with advection. SIAM J. Numer. Anal. 57(5), 2469–2492 (2019)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Lu Zhang

  2. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Daniel Appelö

  3. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Thomas Hagstrom

Authors
  1. Lu Zhang
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  2. Daniel Appelö
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  3. Thomas Hagstrom
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Correspondence to Thomas Hagstrom.

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The authors declare that they have no conflict of interest.

Additional information

This work was supported by NSF Grants DMS-1913076 and DMS-2012296 and completed, while the third author was in residence at the Institute for Computational and Experimental Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Zhang, L., Appelö, D. & Hagstrom, T. Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations. Commun. Appl. Math. Comput. 4, 855–879 (2022). https://doi.org/10.1007/s42967-021-00149-y

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  • DOI: https://doi.org/10.1007/s42967-021-00149-y

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