[en] Perfectly Matched Layers (PMLs) are widely used for the numerical simulation of wave-like problems defined on large or infinite spatial domains. However, for both the time-dependent and the time-harmonic cases, their performance critically depends on the so-called absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyse how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one- and two-dimensional representative time-dependent cases. This study highlights the advantages of the so-called shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions.
Antoine X, Arnold A, Besse C, Ehrhardt M, Schädle A. A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Communications in Computational Physics 2008; 4(4):729-796.
Blayo E, Debreu L. Revisiting open boundary conditions from the point of view of characteristic variables. Ocean Modelling 2005; 9(3):231-252.
Colonius T. Modeling artificial boundary conditions for compressible flow. Annual Review of Fluid Mechanics 2004; 36: 315-345.
Givoli D. 2008. Computational absorbing boundaries. In Computational Acoustics of Noise Propagation in Fluids, chapter 5, Springer, Berlin.
Hagstrom T. Radiation boundary conditions for the numerical simulation of waves. Acta Numerica 1999; 8: 47-106.
Hagstrom T. Radiation boundary conditions for Maxwell's equations: a review of accurate time-domain formulations. Journal of Computational Mathematics 2007; 25: 305-336.
Hagstrom T, Givoli D, Rabinovich D, Bielack J. The double absorbing boundary method. Journal of Computational Physics 2014; 259: 220-241.
Tsynkov S. Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics 1998; 27(4):465-532.
Bérenger J. A perfectly matched layer for the absorption of electromagnetic-waves. Journal of Computational Physics 1994; 114(2):185-200.
Hu F. Development of PML absorbing boundary conditions for computational aeroacoustics: a progress review. Computers & Fluids 2008; 37(4):336-348.
Hu F, Li X, Lin D. Absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations based on the perfectly matched layer technique. Journal of Computational Physics 2008; 227(9):4398-4424.
Abarbanel S, Gottlieb D, Hesthaven J. Non-linear PML equations for time dependent electromagnetics in three dimensions. Journal of Scientific Computing 2006; 28(2-3):125-137.
Bérenger J. Perfectly Matched Layer (PML) for Computational Electromagnetics, Morgan & Claypool: San Rafael, 2007.
Kaufmann T, Sankaran K, Fumeaux C, Vahldieck R. A review of perfectly matched absorbers for the finite-volume time-domain method. Applied Computational Electromagnetics Society Journal 2008; 23(3):184-192.
Petropoulos P. Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates. SIAM Journal on Applied Mathematics 2000; 60(3):1037-1058. Times Cited: 43.
Barucq H, Diaz J, Tlemcani M. New absorbing layers conditions for short water waves. Journal of Computational Physics 2010; 229(1):58-72.
Lavelle J, Thacker W. A pretty good sponge: dealing with open boundaries in limited-area ocean models. Ocean Modelling 2008; 20(3):270-292.
Navon I, Neta B, Hussaini M. A perfectly matched layer approach to the linearized shallow water equations models. Monthly Weather Review 2004; 132(6):1369-1378.
Bécache E, Fauqueux S, Joly P. Stability of perfectly matched layers, group velocities and anisotropic waves. Journal of Computational Physics 2003; 188(2):399-433.
Savadatti S, Guddati M. Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: time harmonic modeling. Journal of Computational Physics 2010; 229(19):6696-6714.
Savadatti S, Guddati M. Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: time-dependent modeling. Journal of Computational Physics 2010; 229(18):6644 -6662.
Nissen A, Kreiss G. An optimized perfectly matched layer for the Schrödinger equation. Communication in Computational Physics 2011; 9: 147-179.
Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R. An exact bounded perfectly matched layer for time-harmonic scattering problems. SIAM Journal on Scientific Computing 2007; 30(1):312-338.
Chew W, Jin J. Perfectly matched layers in the discretized space: an analysis and optimization. Electromagnetics 1996; 16(4):325-340.
Wittwer D, Ziolkowski R. How to design the imperfect berenger PML. Electromagnetics 1996; 16(4):465-485.
Michler C, Demkowicz L, Kurtz J, Pardo D. Improving the performance of perfectly matched layers by means of hp-adaptivity. Numerical Methods for Partial Differential Equations 2007; 23(4):832-858.
Ervedoza S, Zuazua E. Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves. Numerische Mathematik 2008; 109(4):597-634.
Guddati M, Lim K, Zahid M. 2008. Perfectly matched discrete layers for unbounded domain modeling. In Computational Methods for Acoustics Problems, Magoules F (ed), chapter 3, Saxe-Coburg Publications, Stirlingshire, UK.
Gedney S. Perfectly matched layer absorbing boundary conditions. In Computational Electrodynamics: The Finite-Difference Time-Domain Method, Taflove A (ed.). Third edition, Artech House: Boston, 2005; 273-328.
Collino F, Monk P. Optimizing the perfectly matched layer. Computer Methods in Applied Mechanics and Engineering 1998; 164(1-2):157-171.
Datta P, Bhattacharya D. Optimization of uniaxial perfectly matched layer parameters for finite difference time domain simulation and application to coupled microstrip lines with multiple bend discontinuities. International Journal of RF and Microwave Computer-Aided Engineering 2002; 12: 508-519.
Harari I, Albocher U. Studies of FE/PML for exterior problems of time-harmonic elastic waves. Computer Methods in Applied Mechanics and Engineering 2006; 195(29):3854-3879.
Petropoulos P. An analytical study of the discrete perfectly matched layer for the time-domain Maxwell equations in cylindrical coordinates. IEEE Transactions on Antennas and Propagation 2003; 51(7):1671-1675.
Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R. An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. Journal of Computational Physics 2007; 223(2):469-488.
Modave A, Deleersnijder E, Delhez E. On the parameters of absorbing layers for shallow water models. Ocean Dynamics 2010; 60(1):65-79.
Modave A, Kameni A, Lambrechts J, Delhez E, Pichon L, Geuzaine C. An optimum PML for scattering problems in the time domain. The European Physical Journal Applied Physics 2013; 64(11):24502.
Rabinovich D, Givoli D, Bécache E. Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain. International Journal for Numerical Methods in Biomedical Engineering 2010; 26: 1351-1369.
Chew W, Weedon W. A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates. Microwave and Optical Technology Letters 1994; 7(13):599-604.
Petropoulos P. On the termination of the perfectly matched layer with local absorbing boundary conditions. Journal of Computational Physics 1998; 143(2):665-673.
Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R. Perfectly matched layers for time-harmonic second order elliptic problems. Archives of Computational Methods in Engineering 2010; 17(1):77-107.
Harari I, Turkel E. Accurate finite difference methods for time-harmonic wave propagation. Journal of Computational Physics 1995; 119(2):252-270.
Correia D, Jin J-M. On the development of a higher-order PML. IEEE Transactions on Antennas and Propagation 2005; 53(12):4157-4163.
Kuzuoglu M, Mittra R. Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers. IEEE Microwave and Guided Wave Letters 1996; 6(12):447-449.
Roden J, Gedney S. Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media. Microwave and Optical Technology Letters 2000; 27(5):334-339.
Hanert E, Legat V, Deleersnijder E. A comparison of three finite elements to solve the linear shallow water equations. Ocean Modelling 2002; 5: 17-35.
Hughes T, Franca L, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering 1986; 59(1):85-99.
Hesthaven J, Warburton T. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Vol.54, Springer-Verlag: New York, 2008.