Abstract
A gradual long-time growth of the solution in perfectly matched layers (PMLs) has been previously reported in the literature. This undesirable phenomenon may hamper the performance of the layer, which is designed to truncate the computational domain for unsteady wave propagation problems. For unsplit PMLs, prior studies have attributed the growth to the presence of multiple eigenvalues in the amplification matrix of the governing system of differential equations. In the current paper, we analyze the temporal behavior of unsplit PMLs for some commonly used second order explicit finite-difference schemes that approximate the Maxwell’s equations. Our conclusion is that on top of having the PML as a potential source of long-time growth, the type of the layer and the choice of the scheme play a major role in how rapidly this growth may manifest itself and whether or not it manifests itself at all.
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Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)
Bérenger, J.-P.: Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127(2), 363–379 (1996)
Abarbanel, S., Gottlieb, D.: A mathematical analysis of the PML method. J. Comput. Phys. 134(2), 357–363 (1997)
Bécache, E., Joly, P.: On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations. Model. Math. Numer. Anal. 36(1), 87–119 (2002)
Cummer, S.A.: A simple, nearly perfectly matched layer for general electromagnetic media. IEEE Microw. Wirel. Compon. Lett. 13(3), 128–130 (2003)
Kreiss, H.-O., Lorenz, J.: Initial-Boundary Value Problems and the Navier-Stokes Equations. Pure and Applied Mathematics, vol. 136. Academic Press, San Diego (1989)
Gedney, S.D.: An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas Propag. 44(12), 1630–1639 (1996)
Ziolkowski, R.W.: Time-derivative Lorenz material model based absorbing boundary conditions. IEEE Trans. Antennas Propag. 45(10), 1530–1535 (1997)
Abarbanel, S., Gottlieb, D.: On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math. 27(4), 331–340 (1998)
Abarbanel, S., Gottlieb, D., Hesthaven, J.S.: Long time behavior of the perfectly matched layer equations in computational electromagnetics. J. Sci. Comput. 17(14), 405–422 (2002)
Turkel, E., Yefet, A.: Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27, 533–557 (1998)
Bécache, E., Petropoulos, P.G., Gedney, S.D.: On the long-time behavior of unsplit perfectly matched layers. IEEE Trans. Antennas Propag. 52(5), 1335–1342 (2004)
Yee, K.S.: Numerical solution of initial boundary value problem involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307 (1966)
Tsynkov, S.V.: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27, 465–532 (1998)
Qasimov, H., Tsynkov, S.: Lacunae based stabilization of PMLs. J. Comput. Phys. 227, 7322–7345 (2008)
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Research supported by the US Air Force, grant number FA9550-07-1-0170, and US NSF, grant number DMS-0509695.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Abarbanel, S., Qasimov, H. & Tsynkov, S. Long-Time Performance of Unsplit PMLs with Explicit Second Order Schemes. J Sci Comput 41, 1–12 (2009). https://doi.org/10.1007/s10915-009-9282-4
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DOI: https://doi.org/10.1007/s10915-009-9282-4