Optimal PML parameters for efficient numerical simulation of waves in an unbounded domain
Authors: Liu Yijia , Duru Kenneth , Roberts Stephen 
Published: November 2023

The perfectly matched layer (PML) is a perfectly non-reflecting layer that simulates the absorption of waves. However, in practice, once the PML is truncated and discretised, the PML is no longer a completely non-reflecting medium. In this article we discuss how to derive optimal PML parameters for the one dimensional acoustic wave equation. Using a multi-block strategy, we present a numerical implementation of the PML that completely eliminates the PML errors. Numerical experiments are presented to verify the analysis.
References

D. Appelö, T. Hagstrom, and G. Kreiss. Perfectly matched layers for hyperbolic systems: General formulation, well-posedness, and stability. SIAM J. Appl. Math. 67.1 (2006), pp. 1–23. doi: 10.1137/050639107
D. H. Baffet, M. J. Grote, S. Imperiale, and M. Kachanovska. Energy decay and stability of a perfectly matched layer for the wave equation. J. Sci. Comput. 81.3 (2019), pp. 2237–2270. doi: 10.1007/s10915-019-01089-
E. Bécache and M. Kachanovska. Stability and convergence analysis of time-domain perfectly matched layers for the wave equation in waveguides. SIAM J. Numer. Anal. 59.4 (2021), pp. 2004–2039. doi: 10.1137/20M1330543
J.-P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114.2 (1994), pp. 185–200. doi: 10.1006/jcph.1994.1159
A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez. An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J. Comput. Phys. 223.2 (2007), pp. 469–488. doi: 10.1016/j.jcp.2006.09.01
J. Diaz and P. Joly. A time domain analysis of PML models in acoustics. Comput. Meth. Appl. Mech. Eng. 195.29 (2006), pp. 3820–3853. doi: 10.1016/j.cma.2005.02.031
K. Duru. The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38.2 (2016), A1171–A1194. doi: 10.1137/140976443
K. Duru and E. M. Dunham. Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous elastic solids. J. Comput. Phys. 305 (2016), pp. 185–207. doi: 10.1016/j.jcp.2015.10.021
K. Duru, A.-A. Gabriel, and G. Kreiss. On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation. Comput. Meth. Appl. Mech. Eng. 350 (2019), pp. 898–937. doi: 10.1016/j.cma.2019.02.036
K. Duru and G. Kreiss. The perfectly matched layer (PML) for hyperbolic wave propagation problems: A review. arXiv, 2201.03733 (2022). doi: 10.48550/ARXIV.2201.03733
T. Lundquist and J. Nordström. The SBP-SAT technique for initial value problems. J. Comput. Phys. 270 (2014), pp. 86–104. doi: 10.1016/j.jcp.2014.03.048
R. Martin and C. Couder-Castaneda. An improved unsplit and convolutional perfectly matched layer absorbing technique for the Navier–Stokes equations using cut-off frequency shift. Comput. Model. Eng. Sci. 63 (2010), pp. 47–77. doi: 10.3970/cmes.2010.063.04
F. Pled and C. Desceliers. Review and recent developments on the perfectly matched layer (PML) method for the numerical modeling and simulation of elastic wave propagation in unbounded domains. Arch. Comput. Meth. Eng. 29 (2021), pp. 471–518. doi: 10.1007/s11831-021-09581-y
B. Sjögreen and N. A. Petersson. Perfectly matched layer for Maxwell’s equation in second order formulation. J. Comput. Phys. 209 (2005), pp. 19–46. doi: 10.1016/j.jcp.2005.03.01
M. Svärd and J. Nordström. Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268 (2014), pp. 17–38. doi: 10.1016/j.jcp.2014.02.031
E. Vitanza, R. Grammauta, D. Molteni, and M. Monteforte. A shallow water SPH model with PML boundaries. Ocean Eng. 108 (2015), pp. 315–324. doi: 10.1016/j.oceaneng.2015.07.05