Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain

doi:10.1007/s42967-021-00131-8

Abstract

Abstract: We devise hybrid high-order (HHO) methods for the acoustic wave equation in the time domain. We frst consider the second-order formulation in time. Using the Newmark scheme for the temporal discretization, we show that the resulting HHO-Newmark scheme is energy-conservative, and this scheme is also amenable to static condensation at each time step. We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta (SDIRK and ERK) schemes. HHO-SDIRK schemes are amenable to static condensation at each time step. For HHO-ERK schemes, the use of the mixed-order formulation, where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns, is key to beneft from the explicit structure of the scheme. Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(h^k+1) in the H¹-norm and of order O(h^k+2) in the L²-norm. Moreover, test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.

Key words: Hybrid high-order methods, Wave equation, Newmark scheme, Runge-Kutta scheme

CLC Number:

Erik Burman, Omar Duran, Alexandre Ern. Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 597-633.

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References

1.Abbas, M., Ern, A., Pignet, N.:Hybrid high-order methods for fnite deformations of hyperelastic materials.Comput.Mech.62(4), 909-928 (2018)
2.Abbas, M., Ern, A., Pignet, N.:A hybrid high-order method for incremental associative plasticity with small deformations.Comput.Methods Appl.Mech.Eng.346, 891-912 (2019)
3.Angel, J.B., Banks, J.W., Henshaw, W.D.:High-order upwind schemes for the wave equation on overlapping grids:Maxwell's equations in second-order form.J.Comput.Phys.352, 534-567 (2018)
4.Appelö, D., Hagstrom, T.:A new discontinuous Galerkin formulation for wave equations in secondorder form.SIAM J.Numer.Anal.53(6), 2705-2726 (2015)
5.Ayuso de Dios, B., Lipnikov, K., Manzini, G.:The nonconforming virtual element method.ESAIM Math.Model.Numer.Anal.50(3), 879-904 (2016)
6.Banks, J.W., Hagstrom, T., Jacangelo, J.:Galerkin diferences for acoustic and elastic wave equations in two space dimensions.J.Comput.Phys.372, 864-892 (2018)
7.Bécache, E., Joly, P., Tsogka, C.:An analysis of new mixed fnite elements for the approximation of wave propagation problems.SIAM J.Numer.Anal.37(4), 1053-1084 (2000)
8.Botti, L., Di Pietro, D.A., Droniou, J.:A hybrid high-order method for the incompressible NavierStokes equations based on Temam's device.J.Comput.Phys.376, 786-816 (2019)
9.Botti, M., Di Pietro, D.A., Sochala, P.:A hybrid high-order method for nonlinear elasticity.SIAM J.Numer.Anal.55(6), 2687-2717 (2017)
10.Burman, E., Duran, O., Ern, A., Steins, M.:Convergence analysis of hybrid high-order methods for the wave equation.J.Sci.Comput.(2021).https://hal.archives-ouvertes.fr/hal-02922720
11.Burman, E., Ern, A.:An unftted hybrid high-order method for elliptic interface problems.SIAM J.Numer.Anal.56(3), 1525-1546 (2018)
12.Burman, E., Ern, A., Fernández, M.A.:Explicit Runge-Kutta schemes and fnite elements with symmetric stabilization for frst-order linear PDE systems.SIAM J.Numer.Anal.48(6), 2019-2042 (2010)
13.Calo, V., Cicuttin, M., Deng, Q., Ern, A.:Spectral approximation of elliptic operators by the hybrid high-order method.Math.Comput.88(318), 1559-1586 (2019)
14.Cascavita, K.L., Bleyer, J., Chateau, X., Ern, A.:Hybrid discretization methods with adaptive yield surface detection for Bingham pipe fows.J.Sci.Comput.77(3), 1424-1443 (2018)
15.Chave, F., Di Pietro, D., Lemaire, S.:A three-dimensional hybrid high-order method for magnetostatics (2020).https://hal.archives-ouvertes.fr/hal-02407175
16.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)
17.Chung, E.T., Engquist, B.:Optimal discontinuous Galerkin methods for wave propagation.SIAM J.Numer.Anal.44(5), 2131-2158 (2006)
18.Cicuttin, M., Di Pietro, D.A., Ern, A.:Implementation of discontinuous skeletal methods on arbitrarydimensional, polytopal meshes using generic programming.J.Comput.Appl.Math.344, 852-874 (2018)
19.Cockburn, B., Di Pietro, D.A., Ern, A.:Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods.ESAIM Math.Model Numer.Anal.50(3), 635-650 (2016)
20.Cockburn, B., Fu, Z., Hungria, A., Ji, L., Sánchez, M.A., Sayas, F.-J.:Stormer-Numerov HDG methods for acoustic waves.J.Sci.Comput.75(2), 597-624 (2018)
21.Cockburn, B., Gopalakrishnan, J., Lazarov, R.:Unifed hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems.SIAM J.Numer.Anal.47(2), 1319-1365 (2009)
22.Cohen, G., Joly, P., Roberts, J.E., Tordjman, N.:Higher order triangular fnite elements with mass lumping for the wave equation.SIAM J.Numer.Anal.38(6), 2047-2078 (2001)
23.Cohen, G.C.:Higher-Order Numerical Methods for Transient Wave Equations.Springer, Berlin (2002)
24.Di Pietro, D.A., Droniou, J., Manzini, G.:Discontinuous skeletal gradient discretisation methods on polytopal meshes.J.Comput.Phys.355, 397-425 (2018)
25.Di Pietro, D.A., Ern, A.:A hybrid high-order locking-free method for linear elasticity on general meshes.Comput.Methods Appl.Mech.Eng.283, 1-21 (2015)
26.Di Pietro, D.A., Ern, A., Lemaire, S.:An arbitrary-order and compact-stencil discretization of difusion on general meshes based on local reconstruction operators.Comput.Methods Appl.Math.14(4), 461-472 (2014)
27.Di Pietro, D.A., Ern, A., Linke, A., Schieweck, F.:A discontinuous skeletal method for the viscositydependent Stokes problem.Comput.Methods Appl.Mech.Eng.306, 175-195 (2016)
28.Ezziani, A., Joly, P.:Local time stepping and discontinuous Galerkin methods for symmetric frst order hyperbolic systems.J.Comput.Appl.Math.234(6), 1886-1895 (2010)
29.Falk, R.S., Richter, G.R.:Explicit fnite element methods for symmetric hyperbolic equations.SIAM J.Numer.Anal.36(3), 935-952 (1999)
30.Giraldo, F.X., Taylor, M.A.:A diagonal-mass-matrix triangular-spectral-element method based on cubature points.J.Eng.Math.56(3), 307-322 (2006)
31.Griesmaier, R., Monk, P.:Discretization of the wave equation using continuous elements in time and a hybridizable discontinuous Galerkin method in space.J.Sci.Comput.58(2), 472-498 (2014)
32.Grote, M.J., Schneebeli, A., Schötzau, D.:Discontinuous Galerkin fnite element method for the wave equation.SIAM J.Numer.Anal.44(6), 2408-2431 (2006)
33.Hesthaven, J.S., Warburton, T.:Nodal high-order methods on unstructured grids.I.Time-domain solution of Maxwell's equations.J.Comput.Phys.181(1), 186-221 (2002)
34.Krenk, S.:Energy conservation in Newmark based time integration algorithms.Comput.Methods Appl.Mech.Eng.195(44/45/46/47), 6110-6124 (2006)
35.Kronbichler, M., Schoeder, S., Müller, C., Wall, W.A.:Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation.Int.J.Numer.Methods Eng.106(9), 712-739 (2016)
36.Lions, J.-L., Magenes, E.:Non-homogeneous boundary value problems and applications, vols.I, II.Springer, New York (1972) (Translated from the French by P.Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181-182)
37.Marazzato, F., Ern, A., Mariotti, C., Monasse, L.:An explicit pseudo-energy conserving time-integration scheme for Hamiltonian dynamics.Comput.Methods Appl.Mech.Eng.347, 906-927 (2019)
38.Monk, P., Richter, G.R.:A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media.J.Sci.Comput.22(23), 443-477 (2005)
39.Nguyen, N.C., Peraire, J.:Hybridizable discontinuous Galerkin methods for partial diferential equations in continuum mechanics.J.Comput.Phys.231(18), 5955-5988 (2012)
40.Nguyen, N.C., Peraire, J., Cockburn, B.:High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics.J.Comput.Phys.230(10), 3695-3718 (2011)
41.Sánchez, M.A., Ciuca, C., Nguyen, N.C., Peraire, J., Cockburn, B.:Symplectic Hamiltonian HDG methods for wave propagation phenomena.J.Comput.Phys.350, 951-973 (2017)
42.Sjögreen, B., Petersson, N.A.:A fourth order accurate fnite diference scheme for the elastic wave equation in second order formulation.J.Sci.Comput.52(1), 17-48 (2012)
43.Stanglmeier, M., Nguyen, N.C., Peraire, J., Cockburn, B.:An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation.Comput.Methods Appl.Mech.Eng.300, 748-769 (2016)
44.Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.:Polymesher:a general-purpose mesh generator for polygonal elements written in Matlab.Struct.Multidisc.Optim.45, 309-328 (2012)
45.Virta, K., Mattsson, K.:Acoustic wave propagation in complicated geometries and heterogeneous media.J.Sci.Comput.61(1), 90-118 (2014)

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