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153
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
Numerical Solution Of Problems On Unbounded Domains. A Review
 A review, Appl. Numer. Math
, 1998
"... While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many ..."
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Cited by 126 (19 self)
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While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computa...
Radiation Boundary Condition for the Numerical Simulation of Waves
 Acta Numerica
, 1999
"... We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of ..."
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Cited by 91 (3 self)
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We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient timedependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to
1998] The spectral element method: an efficient tool to simulate the seismic response of 2d and 3d geological structures
 Bulletin of Seismological Society of America
"... Abstract We present the spectral element method to simulate lasticwave propagation in realistic geological structures involving complicated freesurface topography and material interfaces for two and threedimensional geometries. The spectral element method introduced here is a highorder variat ..."
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Cited by 87 (7 self)
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Abstract We present the spectral element method to simulate lasticwave propagation in realistic geological structures involving complicated freesurface topography and material interfaces for two and threedimensional geometries. The spectral element method introduced here is a highorder variational method for the spatial approximation of elasticwave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energymomentum conserving scheme that can be put into a classical explicitimplicit predictor/multicorrector format. Longterm energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as At c < 0 (n ~ lind N2), with nel the number of elements, na the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of
On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer
 J. Comput. Phys
, 1995
"... waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative ..."
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Cited by 59 (1 self)
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waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane wave solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing outgoing waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.
Nonreflecting Boundary Conditions For Time Dependent Scattering
 SIAM J. Appl. Math
, 1996
"... An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three space dimensions [1]. Here it is shown how to combine that boundary condition with finite difference methods and finite element methods. Uniqueness of the solution is proved, stabi ..."
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Cited by 55 (2 self)
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An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three space dimensions [1]. Here it is shown how to combine that boundary condition with finite difference methods and finite element methods. Uniqueness of the solution is proved, stability issues are discussed, and improvements are proposed for numerical computation. Numerical examples are presented which demonstrate the improvement in accuracy over standard methods. 1 Supported by an IBM graduate fellowship (grote@cims.nyu.edu). 2 Supported in part by AFOSR, NSF, and ONR (keller@math.stanford.edu). 1 Introduction We wish to calculate numerically the time dependent field u(x; t) scattered from a bounded scattering region in threedimensional space. In this region, there may be one or more scatterers, and the equation for u may have variable coefficients and nonlinear terms. As usual, we surround the scattering region by an artificial boundary B, and confine the comp...
A Formulation of Asymptotic and Exact Boundary Conditions Using Local Operators
 Appl. Num. Math
, 1998
"... In this paper we describe a systematic approach for constructing asymptotic boundary conditions for isotropic wavelike equations using local operators. The conditions take a recursive form with increasing order of accuracy. In three dimensions the recursion terminates and the resulting conditions a ..."
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Cited by 41 (3 self)
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In this paper we describe a systematic approach for constructing asymptotic boundary conditions for isotropic wavelike equations using local operators. The conditions take a recursive form with increasing order of accuracy. In three dimensions the recursion terminates and the resulting conditions are exact for solutions which are described by finite combinations of angular spherical harmonics. First we develop the expansion for the twodimensional wave equation and construct a sequence of easily implementable boundary conditions. We show that in three dimensions the analogous conditions are again easily implementable in addition to being exact. Also, we provide extensions of these ideas to hyperbolic systems. Namely, Maxwell's equations for TM waves are used to demonstrate the construction. Finally, we provide numerical examples to demonstrate the effectiveness of these conditions for a model problem governed by the wave equation. 1 Supported in part by NSF Grant DMS9600146 and by I...
Rapid Evaluation Of Nonreflecting Boundary Kernels For TimeDomain Wave Propagation
 SIAM J. Numer. Anal
, 2000
"... . We present a systematic approach to the computation of exact nonreflecting boundary conditions for the wave equation. In both two and three dimensions, the critical step in our analysis involves convolution with the inverse Laplace transform of the logarithmic derivative of a Hankel function. The ..."
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Cited by 37 (5 self)
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. We present a systematic approach to the computation of exact nonreflecting boundary conditions for the wave equation. In both two and three dimensions, the critical step in our analysis involves convolution with the inverse Laplace transform of the logarithmic derivative of a Hankel function. The main technical result in this paper is that the logarithmic derivative of the Hankel function H (1) # (z) of real order # can be approximated in the upper half z plane with relative error # by a rational function of degree d # O # log # log 1 # +log 2 #+# 1 log 2 1 # # as ###, # # 0, with slightly more complicated bounds for # = 0. If N is the number of points used in the discretization of a cylindrical (circular) boundary in two dimensions, then, assuming that #<1/N , O(N log N log 1 # ) work is required at each time step. This is comparable to the work required for the Fourier transform on the boundary. In three dimensions, the cost is proportional to N...
ON THE ANALYSIS AND CONSTRUCTION OF PERFECTLY MATCHED LAYERS FOR THE LINEARIZED EULER EQUATIONS
, 1997
"... We present a detailed analysis of a recently proposed perfectly matched layer (PML) method for the absorption of acoustic waves. The split set of equations is shown to be only weakly wellposed, and illposed under small low order perturbations. This analysis provides the explanation for the stabil ..."
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Cited by 29 (0 self)
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We present a detailed analysis of a recently proposed perfectly matched layer (PML) method for the absorption of acoustic waves. The split set of equations is shown to be only weakly wellposed, and illposed under small low order perturbations. This analysis provides the explanation for the stability problems associated with the spilt field formulation and illustrates why applying a filter has a stabilizing effect. Utilizing recent results obtained within the context of electromagnetics, we develop strongly wellposed absorbing layers for the linearized Euler equations. The schemes are shown to be perfectly absorbing independent of frequency and angle of incidence of the wave in the case of a nonconvecting mean flow. In the general case of a convecting mean flow, a number of techniques is combined to obtain a absorbing layers exhibiting PMLlike behavior. The efficacy of the proposed absorbing layers is illustrated though computation of benchmark problems in aeroacoustics.