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Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation

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Abstract

We propose a new formulation of explicit time integration for the hybridizable discontinuous Galerkin (HDG) method in the context of the acoustic wave equation based on the arbitrary derivative approach. The method is of arbitrary high order in space and time without restrictions such as the Butcher barrier for Runge–Kutta methods. To maintain the superconvergence property characteristic for HDG spatial discretizations, a special reconstruction step is developed, which is complemented by an adjoint consistency analysis. For a given time step size, this new method is twice as fast compared to a low-storage Runge–Kutta scheme of order four with five stages at polynomial degrees between two and four. Several numerical examples are performed to demonstrate the convergence properties, reveal dispersion and dissipation errors, and show solution behavior in the presence of material discontinuities. Also, we present the combination of local time stepping with h-adaptivity on three-dimensional meshes with curved elements.

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References

  1. Cohen, G.C.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  2. Hesthaven, J.S. Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Application, Vol. 54 of Texts in Applied Mathematics, Springer, Berlin (2008)

  3. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic equations. SIAM J. Numer. Anal. 47(2), 1139–1365 (2009). https://doi.org/10.1137/070706616

    MATH  Google Scholar 

  4. Nguyen, N.C., Peraire, J., Cockburn, B.: High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230, 3695–3718 (2011). https://doi.org/10.1016/j.jcp.2011.01.035

    Article  MathSciNet  MATH  Google Scholar 

  5. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comput. 81(279), 1327–1353 (2011). https://doi.org/10.1090/S0025-5718-2011-02550-0

    Article  MathSciNet  MATH  Google Scholar 

  6. Cockburn, B., Quenneville-Belair, V.: Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation. Math. Comput. 83(285), 65–85 (2013). https://doi.org/10.1090/S0025-5718-2013-02743-3

    Article  MathSciNet  MATH  Google Scholar 

  7. Kronbichler, M., Schoeder, S., Müller, C., Wall, W.: Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. Int. J. Numer. Methods Eng. 106(9), 712–739 (2016). https://doi.org/10.1002/nme.5137

    Article  MathSciNet  MATH  Google Scholar 

  8. Stanglmeier, M., Nguyen, N., Peraire, J., Cockburn, B.: An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Comput. Methods Appl. Mech. Eng. 300, 748–769 (2016). https://doi.org/10.1016/j.cma.2015.12.003

    Article  MathSciNet  Google Scholar 

  9. Kronbichler, M., Kormann, K.: A generic interface for parallel cell-based finite element operator application. Comput. Fluids 63, 135–147 (2012). https://doi.org/10.1016/j.compfluid.2012.04.012

    Article  MathSciNet  MATH  Google Scholar 

  10. Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92 (1980). https://doi.org/10.1016/0021-9991(80)90005-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer, New York (2009)

    Book  MATH  Google Scholar 

  12. Schwartzkopff, T., Munz, C.D., Toro, E.F.: ADER: A high-order approach for linear hyperbolic systems in 2D. J. Sci. Comput. 17(1), 231–240 (2002). https://doi.org/10.1023/A:1015160900410

    Article  MathSciNet  MATH  Google Scholar 

  13. Schwartzkopff, T., Dumbser, M., Munz, C.-D.: Fast high order ADER schemes for linear hyperbolic equations. J. Comput. Phys. 197, 532–539 (2004). https://doi.org/10.1016/j.jcp.2003.12.007

    Article  MATH  Google Scholar 

  14. Dumbser, M., Käser, M.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—II. The three-dimensional isotropic case. Geophys. J. Int. 167, 319–336 (2006). https://doi.org/10.1111/j.1365-246X.2006.03120.x

    Article  Google Scholar 

  15. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016). https://doi.org/10.1016/j.jcp.2016.02.015

    Article  MathSciNet  MATH  Google Scholar 

  16. Dumbser, M., Schwartzkopff, T., Munz, C.-D.: Arbitrary high order finite volume schemes for linear wave propagation. In: Krause, E., Shokin, Y., Resch, M., Shokina, N. (eds.) Computational Science and High Performance Computing II: The 2nd Russian-German Advanced Research Workshop, Stuttgart, Germany, March 14 to 16, 2005, pp. 129–144. Springer, Berlin (2006). https://doi.org/10.1007/3-540-31768-6_11

    Chapter  Google Scholar 

  17. Guo, W., Qiu, J.-M., Qiu, J.: A new Lax–Wendroff discontinuous Galerkin method with superconvergence. J. Sci. Comput. 65(1), 299–326 (2015). https://doi.org/10.1007/s10915-014-9968-0

    Article  MathSciNet  MATH  Google Scholar 

  18. Winters, A.R., Kopriva, D.A.: High-order local time stepping on moving DG spectral element meshes. J. Sci. Comput. 58(1), 176–202 (2014). https://doi.org/10.1007/s10915-013-9730-z

    Article  MathSciNet  MATH  Google Scholar 

  19. Piperno, S.: Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems. ESAIM Math. Model. Numer. Anal. 40(5), 815–841 (2005). https://doi.org/10.1051/m2an:2006035

    Article  MathSciNet  MATH  Google Scholar 

  20. Gassner, G., Hindenlang, G., Munz, C.-D.: A Runge–Kutta based discontinuous Galerkin method with time accurate local time stepping. In: Wang, Z.J. (ed.) Adaptive High-Order Methods in Computational Fluid Dynamics, vol. 2, pp. 95–118. World Scientific Publishing Co. Pte. Ltd., Singapore (2011)

  21. Grote, M., Mehlin, M., Mitkova, T.: Runge–Kutta-based explicit local time-stepping methods for wave propagation. SIAM J. Sci. Comput. 37(2), A747–A775 (2015). https://doi.org/10.1137/140958293

    Article  MathSciNet  MATH  Google Scholar 

  22. Dumbser, M., Käser, M., Toro, E.F.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—V. Local time stepping and \(p\)-adaptivity. Geophys. J. Int. 171, 695–717 (2007). https://doi.org/10.1111/j.1365-246X.2007.03427.x

    Article  Google Scholar 

  23. Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31, 629–651 (1977). https://doi.org/10.1090/S0025-5718-1977-0436612-4

    Article  MathSciNet  MATH  Google Scholar 

  24. Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Math. 35, 177–219 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kubatko, E.J., Yeager, B.A., Ketcheson, D.I.: Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods. J. Sci. Comput. 60, 313–344 (2014). https://doi.org/10.1007/s10915-013-9796-7

    Article  MathSciNet  MATH  Google Scholar 

  26. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods. Wiley, New York (1987)

    MATH  Google Scholar 

  27. Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928). https://doi.org/10.1007/BF01448839

    Article  MathSciNet  MATH  Google Scholar 

  28. Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2005)

    Book  MATH  Google Scholar 

  29. Krivodonova, L., Ruibin, Q.: An analysis of the spectrum of the discontinuous Galerkin method. Appl. Numer. Math. 64, 1–18 (2013). https://doi.org/10.1016/j.apnum.2012.07.008

    Article  MathSciNet  MATH  Google Scholar 

  30. Yakovlev, S., Moxey, D., Kirby, R., Sherwin, S.: To CG or to HDG: a comparative study in 3D. J. Sci. Comput. 67(1), 192–220 (2016). https://doi.org/10.1007/s10915-015-0076-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Hartmann, R.: Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45(6), 2671–2696 (2007). https://doi.org/10.1137/060665117

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, H., Li, F., Qiu, J.: Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput. 55(3), 552–574 (2013). https://doi.org/10.1007/s10915-012-9647-y

    Article  MathSciNet  MATH  Google Scholar 

  33. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004). https://doi.org/10.1016/j.jcp.2004.01.004

    Article  MathSciNet  MATH  Google Scholar 

  34. Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Wells, D.: The deal. II library, version 8.4.0. J. Numer. Math. 24(3), 135–141 (2016). https://doi.org/10.1515/jnma-2016-1045

    Article  MathSciNet  MATH  Google Scholar 

  35. Bangerth, W., Burstedde, C., Heister, T., Kronbichler, M.: Algorithms and data structures for massively parallel generic finite element codes. ACM Trans. Math. Softw. 38(2), 14:1–14:28 (2011). http://dx.doi.org/10.1145/2049673.2049678

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Acknowledgements

The authors acknowledge support by the German Research Foundation (DFG) through the project “High-order discontinuous Galerkin for the exa-scale” (ExaDG) within the priority program “Software for Exascale Computing” (SPPEXA), Grant Agreement Nos. KR4661/2-1 and WA1521/18-1.

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  1. Institute for Computational Mechanics, Technical University of Munich, Boltzmannstr. 15, 85748, Garching b. München, Germany

    Svenja Schoeder, Martin Kronbichler & Wolfgang A. Wall

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  1. Svenja Schoeder
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Correspondence to Svenja Schoeder.

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Schoeder, S., Kronbichler, M. & Wall, W.A. Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation. J Sci Comput 76, 969–1006 (2018). https://doi.org/10.1007/s10915-018-0649-2

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