### Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems

Zhiming Chen, Xinming Wu
<span title="">2012</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cf6hwtzc6baajcjfclnyxmzxgy" style="color: black;">SIAM Journal on Numerical Analysis</a> </i> &nbsp;
The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and the complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML
more &raquo; ... stem for piecewise constant medium property and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium property, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as ln T for large T > 0. Numerical experiments are included to illustrate the efficiency of the PML method. formulations are strongly well-posed, including the uniaxial PML method developed in Sacks et al [25], Zhao-Cangellaris [29], and Gedney [16] for the Maxwell equations and a second order PML formulation in Grote and Sim [17] for wave equations. In curvilinear coordinates, the split-field PML method is introduced in Collino and Monk [13] and the unsplit-field PML methods are introduced in Petropoulous [24] and [26] for Maxwell equations. Although the PML approach has been proved very successful in the practical applications, there are few mathematical results on the convergence of the PML methods. In the frequency domain, the convergence is studied in Lassas and Somersalo [21], Hohage et al [19] for the acoustic scattering problems with the circular layers and in Chen and Wu [10], Kim and Pasciak [20], Chen and Zheng [11] with the uniaxial PML layers. It is proved in [21, 19, 10, 11] that the PML solution converges exponentially to the solution of the original scattering problem as the thickness of the PML layer tends to infinite. In Chen and Wu [9], Chen and Liu [8], an adaptive PML technique is proposed and studied in which a posteriori error estimate is used to determine the PML parameters. For the time-domain PML method, the planar PML method in one space direction is considered in Hagstrom [18] for the wave equation. In de Hoop et al [14], Diaz and Joly [15], the PML system with point source is analyzed based on the Cagniard -de Hoop method. In Chen [7], the convergence of the time-domain PML method with circular layer is proved by using the exponential decay esitmate of the modified Bessel functions. The long-time stability of the PML method is also a much studied topic in the literature (see e.g. Bécache and Joly [3], Bécache et al [4], Appelö et al [2] ). For a PML method to be practically useful, it must be stable in time, that is, the solution should not grow exponentially in time. We remark that the well-posedness of the PML system which follows from the theory of symmetric hyperbolic systems allows the exponential growth of the solutions. In [3, 4, 2, 17] the stability of the Cauchy problem of the PML systems is considered separately in each part of the domain where the PML medium property is assumed to be constant. In this paper we first prove the long-time stability of the initial-boundary value problem of the uniaxial PML system in the PML layer for piecewise constant PML medium properties. The proof is based on an energy argument which is inspired by the method in [3] and the stability analysis of uniaxial PML method in the frequency domain in [11] . We also remark that the stability analysis of the PML method in [20] for the Helmholtz equation is difficult to be used for the time domain analysis as the dependence of the constant in the inf-sup condition on the wave number is not explicit. The second purpose of the paper is to prove the convergence of the time-domain uniaxial PML method. Our technique to prove the PML convergence is different from that for circular PML layer [7] . It is based on the stability estimate for the initialboundary value problem of the PML system in the first part of the paper and an exponential decay estimate of the PML extension in the time domain. This estimate is derived by using the method of Laplace transform, the integral representation of the exterior Dirichlet problem for the Helmholtz equation, and the idea of the complex coordinate stretching. The layout of the paper is as follows. In section 2 we derive the uniaxial PML formulation for (1.1)-(1.4) by using the method of complex coordinate stretching of Chew and Weedon [12] in the frequency domain, and returning to the time domain by using the inverse Laplace transform. In section 3 we show the stability of the initial boundary value problem of the PML system in the PML layer. In section 4 we prove 2
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