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1 Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics S. Abarbanel y, D. Gottlieb z, and J. S. Hesthaven z 1 y Department of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel; z Division of Applied Mathematics, Brown University, Providence, RI 2912, USA We investigate the long time behavior of two unsplit PML methods for the absorption of electromagnetic waves. Computations indicate that both methods suffer from temporal instabilities after the fields reach a quiescent state. The analysis shows that the source of the instabilities is due to the undifferentiated terms of the PML equations and that it is associated with degeneracy of the quiescent systems of equations. This highlights why the instability occurs in special cases only and suggests a remedy to stabilize the PML by removing the degeneracy. Computational results confirm the stability of the modified equations and is used to address the efficacy of the modified schemes for absorbing waves. Key Words: computational electromagnetics, PML, stability 1. INTRODUCTION With the increasing interest in solving Maxwell's equations in the time-domain comes the need to develop accurate, efficient, and robust methods to truncate the computational domain. While this question, relevant to the solution of wave problems in general has received very considerable attention in the past [1, 2, 3, 4, 5, 6] it remains one of the central, yet essentially open, problems. The development of such methods is particularly critical with the increasing use of high-order accurate methods to avoid ruining the accurate interior solution by artificial reflections from the computational boundary. An exciting alternative to these methods was introduced in [7] which proposed the use of an absorbing layer designed in such away that all waves entering the layer, regardless of their frequency and angel of incidence, would be absorbed completely and without reflections into the computational domain. Such layers, termed perfectly matched layers (PML), seemed to overcome the reflection problems and their derivation, first presented for the two-dimensional Maxwell's equations, were 1 Corresponding Author 1

2 2 ABARBANEL, GOTTLIEB, AND HESTHAVEN quickly extended to the three-dimensional problems [8], the equations of acoustics [9], and the equations of linear elasticity [1]. The formulations were all based on a unphysical splitting of the field variables inside the PML region to introduce the additional degrees of freedom needed to design layer with such remarkable properties. However, as was proven in [11], in the case of Maxwell's equations and in [12] for the PML for the linearized Euler equations, the resulting system of PML equations are only weakly wellposed due to the splitting and may become illposed under low-order perturbations. Hence, as was also remarked already in the early work [9], the use of artificial dissipation may be necessary to stabilize the numerical schemes using such formulations. This has generated substantial interest in formulating unsplit PML methods to overcome the stability problems and has resulted in two different formulations, motivated either by physical reasoning [13, 14]or by mathematical arguments [15]. These unsplit PML methods are strongly wellposed, while retaining the properties associated with the original system of PML equations. However, as will be shown in this paper, these two families of wellposed PML methods have problems of their own, i.e., the solutions may diverge in time when the electromagnetic field is quiescent, or nearly quiescent, in the absorbing layer. While the analysis presented in this work provides insight into the source of these problems it does not provide a fully satisfactory solution. It does, however, highlight that much work is still needed in the development of PML methods and sheds light of the need to ensure long time stability of the PML method. The paper is organized as follows. In Sec. 2 we recall the 2D Maxwell's equations and the two types of unsplit PML formulations which forms the basis of the subsequent discussion. Section 3 is devoted to a computational illustration of the stability problems. This will serve as a motivation for the analysis presented in Sec. 4, aimed at highlighting the source of the long time instabilities. Such understanding suggests remedies, which we shall discuss in Sec. 5 where we also illustrate the efficacy of the stabilized PML schemes. Section 6 contains a few concluding remarks. 2. MAWELL'S EQUATIONS AND PML METHODS We shall consider the normalized two-dimensional Maxwell's equations in the TE polarized z ; ; ; where (E x ;E y ) represents the electric field in the plane and H z is the magnetic field component, oriented out of the plane. For simplicity and without loss of generality, Eq.(1) is normalized such that the permittivity and permeability are unity and all lengths are in terms of a reference length scale.

3 LONG TIME BEHAVIOR OF PML METHODS IN CEM 3 Solving Eq.(1) in an open domain requires attention to the truncation of the computational domain. For this purpose we shall consider two methods, both being perfectly matched layer methods. The basic idea for such techniques is to surround the computational domain with an extra layer in which adifferent set of equations are being solved. These equations reduce to the original Maxwell's equations on the layer-vacuum interface, ensuring the matching, yet waves entering the layer are damped regardless of the frequency and angle of incidence. The perfectly matched layer property is obtained by introducing additional degrees of freedom in the absorbing layer equations. Not surprisingly, however, this can be done in various different ways, all leading to layers with similar properties but with subtle differences. It is the purpose of this paper to analyze the temporal instabilities which arise under special conditions of quiescence within the two types of layers and to highlight some of the differences between the two formulations. We assume throughout that the layer is in the direction of the x-axis Physically Motivated PML Method By considering the absorbing layer to have certain physical properties, e.g., as a Lorenz material [14] or as an anisotropic material [13], one recovers schemes that can be written as + ff(e x P ) ; (2) ffe y ; (3) x ffh z ; (4) = ff(e x P ) : (5) We shall specify the absorption coefficient, ff = ff(x), shortly. It is worthwhile to note that the auxiliary variable, P, is satisfied by an ordinary differential equation only and, thus, adds very little additional computational work Mathematically Derived PML Method Taking a different approach by seeking systems of equations, supporting exponentially decaying waves at all angles of incidence and frequencies, and reducing to Maxwell's equations at the vacuum-layer interface, one obtains a set of layer equation, see z ; (6) 2ffE y ffp ; (7) x + ff Q ; (8)

4 4 ABARBANEL, GOTTLIEB, AND = ffe y ; = ffq E y : (1) Similar to the previous PML, Eqs.(2)-(5), the additional field variables, P and Q, satisfy ordinary differential equations, although in this case two rather than one. However, as shown in [15], this ensures uniform decay within the absorbing layer contrary to the former formulation which may allow nonuniform decay with small regions of growth close to the vacuum-layer interface. 3. COMPUTATIONAL MOTIVATION FOR ANALSIS To highlight problems with the two PML methods introduced above, we shall consider a very simple test case. The computational domain consists of a rectangular two-dimensional domain with (x; y) 2 [ 5; 5] 2 as illustrated in Fig. 1. To terminate the computational domain in x-direction, we add two additional layers, having 5» jxj» 6, in which the PML equations are solved. To terminate the domain in the y-direction we use simple characteristic boundary conditions. Other choices have been made for this latter termination without changing the results in any qualitative way Perfectly Matched Layer Computational Domain Perfectly Matched Layer y x FIG. 1. The geometry of the test case used throughout. As the initial condition we use a magnetic pulse of the form H z (x; y; ) = exp» (ln 2) x2 + y 2 In all computations ffi = 3 and the initial electric field components are taken to be zero. To solve Maxwell's equations and the PML equations, we use a 4th order explicit finite-difference method and an explicit 4th order Runge-Kutta method in time. The three computational regions are connected by the use of characteristic variables in a classical multi-domain fashion and 3rd order one-sided stencils are used to terminate the stencil at the boundary of the domain. No artificial dissipation is ffi 2 :

5 LONG TIME BEHAVIOR OF PML METHODS IN CEM 5 a) b) c) d) e) FIG. 2. Short time dynamics of Maxwell's equations subject to a magnetic initial pulse. The snapshots are given at T= (a), T=2 (b), T=4 (c), T=6 (d), and T=8(e), with left column showing E x, the middle column shown E y, and the right column H z. All figures are with contours between -.1 and.1 at intervals of :1, exempting the zero contour. used. Except otherwise stated we use an equidistant grid everywhere with x = y = 1, i.e., there is about 1 points in the initial pulse. The short time behavior of the computation is illustrated in Fig. 2, showing how the initial pulse spreads, enters the PML and is being effectively absorbed. In this particular case we have used the PML in Eqs.(2)-(5), but the alternative in Sec. 2.2 yields almost equivalent results. Here and in all subsequent computations we use an absorption profile in the PML given as

6 6 ABARBANEL, GOTTLIEB, AND HESTHAVEN a) b) c) d) e) FIG. 3. Longtime dynamics of Maxwell's equations subject to a magnetic initial pulse and with Eqs.(2)-(5) as the PML and x = y =1:. The snapshots are given at T=1 (a), T=2 (b), T=3 (c), T=4 (d), and T=5 (e). The left column shows E x, the middle column shown E y, and the right columnh z. All figures are with contours between -.1 and.1 at intervals of :1, exempting the zero contour. ff(x) = ( m jxj 5 1 5»jxj»6 jxj»5 ; (11) where m is the order of the profile. Typical values are m =2 4andwe shall use m = 3 unless otherwise stated.

7 LONG TIME BEHAVIOR OF PML METHODS IN CEM 7 To appreciate the differences between the two PML methods we need to consider the long-time behavior of the two formulations. As a first example of the longtime behavior of the PML schemes, we illustrate in Fig.3 the results obtained when extending the simulation in Fig. 2 to later times. The PML used is the physically motivated one given in Eqs.(2)-(5). Unexpectedly, we find that long time after the initial pulse has left the computational domain, the PML layer exhibits growth which eventually enters the computational domain and impacts the solution. We note that the growth is most pronounced in the E x component while the errors in the E y and H z components seem to be driven by the former. Furthermore, the growth seems to be evident throughout the full width of the layer. a) b) c) FIG. 4. Longtime dynamics of Maxwell's equations subject to a magnetic initial pulse and with Eqs.(2)-(5) as the PML and x = y =:8. The snapshots are given at T=3 (a), T=5 (b), and T=7 (c). The left column shows E x, the middle column shown E y, and the right column H z. All figures are with contours between -.1 and.1 at intervalsof:1, exempting the zero contour. To gain a better understanding of the nature of this instability, we repeat the experiment above using a refined grid with x = y = :8. Snapshots of the solution are shown in Fig.4 and we observe growth similar to that seen in Fig. 3. However, delay in the growth as compared to results on the coarser grid seems to indicate that the level of truncation errors in the scheme affects the results. Let us finally repeat the experiment with the mathematically derived PML, Eqs.(6)-(1), and x = y = 1: as for the results in Fig. 3. The results are shown in Fig. 5, confirming that this formulation also suffers from long-time stability problems inside the layers. Computations confirm that increasing the resolution postpones the onset of the problem but does not remove it. Comparing with the re-

8 8 ABARBANEL, GOTTLIEB, AND HESTHAVEN sults in Fig. 3 a faster growth is noteworthy as well as a more significant localization of the growth region close to the vacuum/pml interface. a) b) c) d) FIG. 5. Longtime dynamics of Maxwell's equations subject to a magnetic initial pulse and with Eqs.(6)-(1) as the PML. The snapshots are given at T=1 (a), T=14 (b), T=18 (c), and T=22 (d). The left column shows E x, the middle column shown E y, and the right column H z. All figures are with contours between -.1 and.1 at intervals of :1, exempting the zero contour. 4. ANALSIS OF THE PML EQUATIONS The central observations to draw from the experiments in Sec. 3 can be summarized in the following way ffl Both the physically motivated PML, Eqs.(2)-(5), and the mathematically derived PML, Eqs.(6)-(1), appear to suffer from temporal stability problems long after the pulse has passed through the PML, i.e., the solution is essentially constant in space. ffl In Eqs.(2)-(5) E x appears to be the driving force, while E x and/or E y seems to be the most rapidly growing components in Eqs.(6)-(1).

9 LONG TIME BEHAVIOR OF PML METHODS IN CEM 9 ffl The growth in Eqs.(6)-(1) is faster than in Eqs.(2)-(5) and appears to be most pronounced around the vacuum/pml interface. ffl The instability seems to be sensitive to the level of truncation error, i.e., a smaller truncation error postpones the instability but does not remove it. In the following section we shall provide an analysis of the two PML methods with the aim of explaining the observed behavior and the differences between them in order to suggest ways to resolve the temporal instability issue. For simplicity we present the analysis of the two formulations separately although, as we shall see, the source of the problems in the two formulations is closely related Analysis of the Physically Motivated PML Equations The starting point of our analysis is the observation that the temporal growth in the numerical solution starts when the absorbing layer is almost quiescent. From Eqs.(2)-(5) we note that = the equations for E x and P decouple and we Ex P = ff»» 1 1 Ex 1 1 P = ffmq ; (12) where q =[E x ;P] T and the coefficient matrix, M, of the initial value problem is M=» : (13) The coefficient matrix is clearly singular with a double eigenvalue, (M) =, i.e., M 2 = and the solution to Eq.(12) takes the form q =exp(ffmt) q = q + fftmq ; (14) where q =[E x ;P ] T represents the values of E x and P once the field is quiescent at which point t, whichisvery large in the original scenario, is reset to be zero. Note that Eq.(14) confirms that E x can be expected to the physical quantity where the impact of the instability is seen first and that one should expect the truncation error, i.e., the value of q, to impact the growth rate. Finally, the linear growth can be expected to vary inside the layer and happen everywhere inside the layer. This behavior is consistent with the results shown in Figs In contradistinction to this result, derived under the assumption =, consider the case 6=,while@=@x =. Assuming periodicity in y and i! y into Eqs.(2)-(5) yields a decoupled system for E x, H z, and P where q =[E x ;H z ;P] T = ~Mq ;

10 1 ABARBANEL, GOTTLIEB, AND HESTHAVEN 2 3 ff i! ff ~M = 6 i! ff : ff ff As the eigenvalues are ( ~M) = ff and ( ~M) = ±i!, and, from Eq.(3), E y is uniformly decaying, the PML is stable. In other words, any variation in y will stabilize the physically motivated PML Analysis of the Mathematically Derived PML Equations Approaching the analysis as above, we assume that the solution in the PML layer is quiescent and neglect all spatial derivatives. This implies that Eqs.(6)-(1) take the form =Mq ; (15) where q =[E x ;E y ;H z ;P;Q] T and the coefficient matrixis Observe thate y and P decouple as 2 3 2ff ff M= 6 ff 7 4 ff 5 : (16) 1 Ey P = ff» 2 1 1» Ey P = ffmq ; (17) where q =[E y ;P] T and the coefficient matrix, M, of the initial value problem is M=» As (M) = 1 of multiplicity two is degenerate, the solution to Eq.(17) takes the form q = q e fft + ffte fft» : q ; (18) where q =[E y ;P ] T represents the values of E y and P after the PML layer has reached the quiescent state.

11 LONG TIME BEHAVIOR OF PML METHODS IN CEM 11 Notice that while E y and P exhibit an initial growth in time, they remain bounded with for all values of t and ff by jp j max =(P + E y )e Ey Ey +P ; je y j max = e 1 jp j max : Substituting the solution in Eq.(18) into Eqs.(15)-(16) we recover Q = e fft» Q E y t (E y + P )fft 2 ; (19) H z = K + e fft [A + Bt + Ct 2 ] ; where K = H ff ff Q + ff ff 2 P ; A = ff ff 2 P ff B = ff ff P ; C = 1 2 ff (E y + P ) : ff Q ; (2) This result shows that the worst possible scenario results from the triple multiplicity of (M) = ff in Eq.(16), as these eigenvalues have only one eigenvector associated with them, leading to terms like t 2 e fft. While H z and Q will both decay asymptotically in time, they attain arbitrarily large values for sufficiently small values of ff. This is fully consistent with the observations made in Sec. 3 in general. In particular this analysis explains why the instability is predominantly localized to the vacuum-pml interface where ff is small and why the growth in the mathematically derived PML, Eqs.(6)-(1), can be expected to be faster than that observed in the physically motivated PML, Eqs.(2)-(5). 5. STABILIZING THE LONG TIME BEHAVIOR With this understanding of the source of the stability problems observed in Sec. 3wearenow in a position to consider ways of stabilizing the PML equations. In the following we shall discuss ways of doing so, concluding with a few numerical experiments to verify how well these stabilized PML equations absorb waves Stabilizing the Physically Motivated PML Equations From the discussion in Sec. 4.1 it is clear that the source of the linear growth in the PML equations, Eqs.(2)-(5), is the double degenerate zero eigenvalue of the coefficient matrix, M, given in Eq.(13). This suggests a cure to the problem by perturbing M to split this double root into two distinct eigenvalues with negative real part. Here we choose to modify the PML equations, Eqs.(2)-(5), as y + ff((1 ")E x P ) ; (21) ffe y ; (22)

12 12 ABARBANEL, GOTTLIEB, AND x ffh z ; (23) = ff(e x P ) : (24) We note that in the case of a quiescent state, the equations for E x and P still decouple, but the coefficient matrix,m ", for this modified system is " 1 M " = ff 4 5 : 1 1 The eigenvalues, now clearly distinct for " 6=, are given by (M " )= ff " " 2 ± r # " 2 " 2 and have negative real parts for all ">. It is not clear, a priori, that the new set of equations, Eqs.(21)-(24), defines an absorbing layer. Clearly, taking " fi 1 will not perturb the original PML equations too severely and one can expect the two formulations to behave similarly. It is not clear, however, whether the modified equations strictly speaking will be a PML for any finite value of ". We shall return to this critical question, by means of some numerical experimentation in Sec Stabilizing the Mathematically Derived PML Equations Stabilizing the mathematically derived PML, Eqs.(6)-(1), can be done in several ways and some are quite likely better than others. By inspection of Eqs.(19)-(2) it is evident that taking ff = would prevent the instability. This naturally suggests that using a ff = constant profile is stable and numerical experiments confirm this. However, taking ff = constant causes severe reflections from the vacuum/pml interface as the PML no longer is perfectly matched, i.e., this is not a feasible option. Alternatively, one could simply neglect the ff Q term in Eq.(8) while still using for ff avariable profile as in Eq.(11) in the remaining equations. One should expect this to be stable, as has also be confirmed through computations. However, the inconsistent specification of ff and ff throughout Eqs.(6)-(1) may well destroy the absorption characteristics of the PML, hence reducing the value of the scheme. We shall return to this shortly. Prior to that, however, let us approach the problem in a way similar to that used to cure the problems in Eqs.(2)-(5). Among several alternatives, we choose to modify Eqs.(6)-(1) as follows z ; (25) 2ffE y ffp 2μE y ; (26) x + ff Q ; (27)

13 a)1-2 b)1-2 LONG TIME BEHAVIOR OF PML METHODS IN CEM = ffe y ; = ffq E y ; (29) i.e., only Eq.(26) is modified. As in the case of the original system of equations, Eq.(26) and Eq.(28), decouple in the quiescent state. The coefficient matrix, M μ, for this modified system is M μ = The eigenvalues of this operator are» 2ff 2μ ff ff : (M μ )= (ff + μ) ± p (ff + μ) 2 ff 2 : Clearly, as long as μ 6= and μ> ff, the eigenvalues are distinct with negative real parts, i.e., the scheme is stable. Note that as was the case for Eqs.(21)-(24), we are unable to verify analytically that Eqs.(25)-(29) maintains the spatial decay property of the original PML equations Numerical Verification Numerical experiments have verified that the modified PML equations, Eqs.(21)- (24) and Eqs.(25)-(29), indeed do not display the temporal instabilities found in the original PML formulations. However, as we can not answer the question of whether the time-stabilized equations retain the spatial decays properties of the original PML equations, we shall study this important aspect via numerical experimentation. 1-3 Original PML 1-3 Original PML E y -E y PML Stabilized PML H z -H z PML Stabilized PML t FIG. 6. Reflections from the physically motivated PML, Eqs.(2)-(5), and its stabilized version, Eqs.(21)-(24). In a) we showthel2 reflection errors for E y and in b) for H z. In both cases, (E y;h z) represents the results obtained in a large domain and (E PML y ;H PML z ) are computed with the PML. t To compare and evaluate the different methods we shall use the example studied in Sec. 3. While the problem has an analytic solution [16], we shall use one computed in a sufficiently large domain as the reference solution. This allows for a clear separation between errors caused by the discretization and errors caused

14 a)1-2 b) ABARBANEL, GOTTLIEB, AND HESTHAVEN by reflections from the absorbing layers. We shall measure the time-dependent reflection error in L 2 along a line, parallel to the vacuum/pml interface, 2 grid cells inside the computational domain. In Fig. 6weshow the reflection errors computed using the physically motivated PML in its original form, Eqs.(2)-(5), as well as in its stabilized form, Eqs.(21)-(24). In the latter weuse" =1=3which has proven to suffice to stabilize the PML scheme for very long time with no signs of growth. It should be remarked that little effort has been made to optimize this value which maywell depend on the problem as well as on the details of the absorption profile, Eq.(11). Nevertheless, for both E y and H z the stabilized PML performs as well as the original PML. Similar results have been obtained for a number of other test cases leading us to conjecture that the stabilized form, Eqs.(21)-(24), retains the absorption properties of a PML. Whether this is true in a strict sense for general problems remains unknown. Let us now consider the mathematically derived PML which turns out to be more problematic. As the results in Sec. 3 and the analysis in Sec. 4.2 showed, it is for small values of ff that the instability is most pronounced, i.e., where the stabilization is most needed. This indicates that μ in Eqs.(25)-(29) must dominate in the region close to the PML interface to stabilize the scheme. In the subsequent experiments we have used μ = :4jjxj 5j, i.e., a matched linear profile. This was found stabilize the scheme for very long time, although no effort was made to optimize it for various ff and different problems. σ = PML σ = PML E y -E y PML Stabilized PML Original PML H z -H z PML Stabilized PML Original PML t FIG. 7. Reflections from the mathematically motivated PML, Eqs.(6)-(1), and its stabilized version, Eqs.(25)-(29). In a) we showthel2 reflection errors for E y and in b) for H z. In both cases, (E y;h z) represents the results obtained in a large domain and (E PML y ;H PML z ) are computed with the PML. t Figure 7 shows the reflection errors using the original PML, Eqs.(6)-(1) and the stabilized PML, Eqs.(25)-(29), with μ as discussed above. We also show the results computed with Eqs.(6)-(1) by taking ff as in Eq.(11) but making the inconsistent assumption that ff =. As the analysis in Sec. 5.2 showed, this would stabilized the scheme and the computational experiments confirm that. From the results it is clear that the absorbing properties of the mathematically derived PML equations are very sensitive to modifications of any kind, increasing the reflections with up to two orders of magnitude over the original PML. While the inconsistent ff = modification seems worst, the performance of Eqs.(25)-(29) is very similar and inferior to the stabilized version of the physically motivated PML, Eqs.(21)-(24).

15 LONG TIME BEHAVIOR OF PML METHODS IN CEM 15 This leads us to conclude that the modifications needed to stabilized Eqs.(6)-(1) seem to diminish the absorbing properties. It should be noted, however, that for short times or cases where the instability causes no problem, Eqs.(6)-(1) is superior to Eqs.(2)-(5). 6. CONCLUDING REMARKS While unsplit PML methods do not suffer from instabilities due to loss of strong wellposedness as does the original split PML scheme, wehaveshown here that these methods may be susceptible to other kinds of instabilities. The stability problems are associated with the undifferentiated terms in the PML equations and manifest themselves through slowly growing solutions inside a quiescence layer. Although the growth is slow, the computational results confirm that they will eventually spread and pollute the computational domain. The understanding of the source of the instability also suggested ways of overcoming the problems. However, how these slight modifications of the PML methods alter their performance as an absorbing layer remains an open question. Indeed, the computational results suggest that the impact may depend strongly on the details of the PML formulation. As the need to do very long time computations using high-order schemes becomes more and more apparent, the results presented here suggests that continued developments of PML methods are needed. In particular, it seems natural that one should also consider the long time stability, together with the traditional PML properties, as part of the construction. We hope to be able to report on such developments in future work. ACKNOWLEDGMENT The authors acknowledge partial support for this work from AFOSR/DARPA under contract F The third author (JSH) furthermore acknowledge partial support from NSF under contract DMS-74257, from ARO under contract ARO-4464-MA, and from the Alfred P. Sloan Foundation as a Sloan Research Fellow. REFERENCES 1. B. Engquist and A. Majda, Absorbing Boundary Conditions for the Numerical Solution of Waves, Math. Comp. 31(1977), pp A. Bayliss and E. Turkel, Radiation Boundary Conditions for Wave-Like Equations, Comm. Pure Appl. Math. 23(198), pp D. Givoli, Nonreflecting Boundary Conditions, J. Comput. Phys. 94(1991), pp M. Grote and J. Keller, Nonreflecting Boundary Conditions for Time Dependent Scattering, J. Comput. Phys. 127(1996), pp S. V. Tsynkov, Numerical Solution of Problems on Unbounded Domains, Appl. Numer. Math. 27(1998), pp T. Hagstrom, Radiation Boundary Conditions for the Numerical Simulation of Waves, Acta Numerica 8(1999), pp J. P. Berenger, A Perfectly Matched Layer for the Absorption of Electromagnetic Waves, J. Comput. Phys. 114(1994), pp J.. Berenger, Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves, J. Comput. Phys. 127(1996), pp F. Q. Hu, On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer, J. Comput. Phys. 129(1996), pp

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