×

Energy decay and stability of a perfectly matched layer for the wave equation. (English) Zbl 1447.35304

The authors develop the perfectly matched layer approach proposed in the papers [M. J. Grote and I. Sim, “Efficient PML for the wave equation”, Preprint, arXiv:1001.0319; “Perfectly matched layer for the second-order wave equation”, in: Proceedings of the ninth international conference on numerical aspects of wave propagation, WAVES 2009. 370–371 (2009)]. In the present paper, the authors consider the wave equation in its standard second-order matrix form in a domain surrounded by a perfectly matched layer. On the outer boundary the homogeneous Dirichlet or Neumann conditions are imposed. The authors prove that the corresponding energy functional in \(L_2\) is a nonincreasing function of time for continuous and discrete formulations with constant damping coefficients. Numerical results validate the theory.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
35B35 Stability in context of PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Abarbanel, S.; Gottlieb, D., A mathematical analysis of the PML method, J. Comput. Phys., 134, 357-363 (1997) · Zbl 0887.65122 · doi:10.1006/jcph.1997.5717
[2] Abarbanel, S.; Gottlieb, D.; Hestahaven, JS, Well-posed perfectly matched layers for advective acoustics, J. Comput. Phys., 154, 266-283 (1999) · Zbl 0947.76076 · doi:10.1006/jcph.1999.6313
[3] Appelö, D.; Hagstrom, T.; Kreiss, G., Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math., 67, 1-23 (2006) · Zbl 1110.35042 · doi:10.1137/050639107
[4] Appelö, D.; Kreiss, G., Application of a perfectly matched layer to the nonlinear wave equation, Wave Motion, 44, 531-548 (2007) · Zbl 1231.65186 · doi:10.1016/j.wavemoti.2007.01.004
[5] Barucq, H.; Diaz, J.; Tlemcani, M., New absorbing layers conditions for short water waves, J. Comput. Phys., 229, 58-72 (2010) · Zbl 1381.76037 · doi:10.1016/j.jcp.2009.08.033
[6] Bécache, E.; Fauqueux, S.; Joly, P., Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys., 188, 399-433 (2003) · Zbl 1127.74335 · doi:10.1016/S0021-9991(03)00184-0
[7] Bécache, E.; Joly, P., On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations, M2AN Math. Model. Numer. Anal., 36, 87-119 (2002) · Zbl 0992.78032 · doi:10.1051/m2an:2002004
[8] Bécache, E.; Joly, P.; Kachanovska, M., Stable perfectly matched layers for a cold plasma in a strong background magnetic field, J. Comput. Phys., 341, 76-101 (2017) · Zbl 1376.78003 · doi:10.1016/j.jcp.2017.03.051
[9] Bécache, É., Joly, P., Kachanovska, M., Vinoles, V.: Perfectly matched layers in negative index metamaterials and plasmas. CANUM 2014-42e Congrès National d’Analyse Numérique, ESAIM Proc. Surveys, vol. 50, pp. 113-132. EDP Sci, Les Ulis (2015) · Zbl 1341.78013 · doi:10.1051/proc/201550006
[10] Bécache, E.; Joly, P.; Vinoles, V., On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials, Math. Comput., 87, 2775-2810 (2018) · Zbl 1404.35427 · doi:10.1090/mcom/3307
[11] Bécache, E.; Kachanovska, M., Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: necessary and sufficient conditions of stability, ESAIM Math. Model. Numer. Anal., 51, 2399-2434 (2017) · Zbl 1454.78010 · doi:10.1051/m2an/2017019
[12] Bécache, E.; Petropoulos, PG; Gedney, SD, On the long-time behavior of unsplit perfectly matched layers, IEEE Trans. Antennas Propag., 52, 1335-1342 (2004) · Zbl 1368.78159 · doi:10.1109/TAP.2004.827253
[13] Bérenger, JP, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 185-200 (1994) · Zbl 0814.65129 · doi:10.1006/jcph.1994.1159
[14] Bérenger, JP, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 127, 363-379 (1996) · Zbl 0862.65080 · doi:10.1006/jcph.1996.0181
[15] Chabassier, J.; Imperiale, S., Space/time convergence analysis of a class of conservative schemes for linear wave equations, C. R. Math. Acad. Sci. Paris, 355, 282-289 (2017) · Zbl 1360.65227 · doi:10.1016/j.crma.2016.12.009
[16] Chen, Z., Convergence of the time-domain perfectly matched layer method for acoustic scattering problems, Int. J. Numer. Anal. Model., 6, 124-146 (2009) · Zbl 1158.76394
[17] Chew, WC; Weedon, WH, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microw. Opt. Technol. Lett., 7, 599-604 (1994) · doi:10.1002/mop.4650071304
[18] Collino, F.; Monk, P. B., Optimizing the perfectly matched layer, Computer Methods in Applied Mechanics and Engineering, 164, 157-171 (1998) · Zbl 1040.78524 · doi:10.1016/S0045-7825(98)00052-8
[19] Collino, F.; Tsogka, C., Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 66, 294-307 (2001) · doi:10.1190/1.1444908
[20] Demaldent, E.; Imperiale, S., Perfectly matched transmission problem with absorbing layers: application to anisotropic acoustics in convex polygonal domains, Int. J. Numer. Methods Eng., 96, 689-711 (2013) · Zbl 1352.74342 · doi:10.1002/nme.4572
[21] Diaz, J.; Joly, P., A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Eng., 195, 3820-3853 (2006) · Zbl 1119.76046 · doi:10.1016/j.cma.2005.02.031
[22] Duru, K., The role of numerical boundary procedures in the stability of perfectly matched layers, SIAM J. Sci. Comput., 38, a1171-a1194 (2016) · Zbl 1339.65112 · doi:10.1137/140976443
[23] Duru, K.; Kreiss, G., Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form, SIAM Numer. Anal., 52, 2883-2904 (2014) · Zbl 1311.35138 · doi:10.1137/13093563X
[24] Ervedoza, S.; Zuazua, E., Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves, Numer. Math., 109, 597-634 (2008) · Zbl 1148.65070 · doi:10.1007/s00211-008-0153-y
[25] Grote, M., Sim, I.: Efficient PML for the wave equation. Preprint. arXiv:1001.0319 [math:NA] (2010)
[26] Grote, M.J., Sim, I.: Perfectly matched layer for the second-order wave equation. In: Proceedings of the Ninth International Conference on Numerical Aspects of Wave Propagation (WAVES 2009, held in Pau, France, 2009), pp. 370-371
[27] Hagstrom, T.; Appelö, D., Automatic symmetrization and energy estimates using local operators for partial differential equations, Commun. Partial Differ. Equ., 32, 1129-1145 (2007) · Zbl 1124.35013 · doi:10.1080/03605300600854258
[28] Hu, FQ, On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer, J. Comput. Phys., 129, 201-219 (1996) · Zbl 0879.76084 · doi:10.1006/jcph.1996.0244
[29] Hu, FQ, A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, J. Comput. Phys., 173, 455-480 (2001) · Zbl 1051.76593 · doi:10.1006/jcph.2001.6887
[30] Joly, P., An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation, SeMA J., 57, 5-48 (2012) · Zbl 1331.35003 · doi:10.1007/BF03322599
[31] Kachanovska, M.: Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: energy estimates (2017). https://hal.inria.fr/hal-01419682. Https://hal.inria.fr/hal-01419682 · Zbl 1454.78010
[32] Kaltenbacher, B.; Kaltenbacher, M.; Sim, I., A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics, J. Comput. Phys., 235, 407-422 (2013) · Zbl 1291.35122 · doi:10.1016/j.jcp.2012.10.016
[33] Komatitsch, D.; Tromp, J., A perfectly matched layer absorbing boundary conditionfor the second-order seismic wave equation, Geophys. J. Int., 154, 146-153 (2003) · doi:10.1046/j.1365-246X.2003.01950.x
[34] Nataf, F., A new approach to perfectly matched layers for the linearized Euler system, J. Comput. Phys., 214, 757-772 (2006) · Zbl 1088.76052 · doi:10.1016/j.jcp.2005.10.014
[35] Sjögreen, B.; Petersson, NA, Perfectly matched layers for Maxwell’s equations in second order formulation, J. Comput. Phys., 209, 19-46 (2005) · Zbl 1073.78014 · doi:10.1016/j.jcp.2005.03.011
[36] Zhao, L.; Cangellaris, AC, Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids, IEEE Trans. Microw. Theory Tech., 44, 2555-2563 (1996) · doi:10.1109/22.554601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.