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55
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
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...
Nonreflecting Boundary Conditions for the TimeDependent Wave Equation
 J. Comput. Phys
, 2002
"... this paper, we couple fast nonreflecting boundary conditions, developed in [3] for spherical and cylindrical boundaries and here for planar boundaries, to finitedifference solvers for the wave equation. In Section 2, we describe the exact (nonlocal) formulation, and in Section 3 we develop the fast ..."
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Cited by 27 (3 self)
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this paper, we couple fast nonreflecting boundary conditions, developed in [3] for spherical and cylindrical boundaries and here for planar boundaries, to finitedifference solvers for the wave equation. In Section 2, we describe the exact (nonlocal) formulation, and in Section 3 we develop the fast algorithm for handling the convolution operators that arise. In Section 4, we present simple temporal and spatial discretization schemes, and in Section 5, we present a number of numerical experiments. We compare the performance of our exact scheme, local EngquistMajda conditions [10], and the recently popular PML method [7], which uses an absorbing region to dampen undesired reflections. Our conclusions and directions for future work are discussed in Section 6
Highorder nonreflecting boundary scheme for timedependent waves
 Journal of Computational Physics
"... waves ..."
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Highorder nonreflecting boundary conditions for dispersive waves
, 2003
"... Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the fin ..."
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Cited by 24 (5 self)
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Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the finite domain bounded by B. The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original loworder implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdontype NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via numerical examples.
Artificial Boundary Conditions of Absolute Transparency for 2D and 3D External TimeDependent Scattering Problems
 Eur. J. Appl. Math. 9
, 1996
"... this paper is to make accessible the results of preprints [12], [13]. Besides, we shall show that the conditions [10] and [14] are equivalent. Finally, we give the results of test calculations for 2D and 3D cases: the formulation of the first test problem is from [15]; the second test problem corres ..."
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Cited by 22 (1 self)
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this paper is to make accessible the results of preprints [12], [13]. Besides, we shall show that the conditions [10] and [14] are equivalent. Finally, we give the results of test calculations for 2D and 3D cases: the formulation of the first test problem is from [15]; the second test problem corresponds to one of 2D benchmarks from [22]. The calculation of a number of scattering problems presented in [15] and here demonstrate a high superiority of exact ABCs based on the Fourier method for spherical and polar grids. In Section 5, we propose a way of treating the artificial boundary of a nonspherical shape; the numerical investigation of our conditions coupled with Cartesian mesh in the computational domain is planned. 2. Problem Formulation Consider in IR
HighOrder RKDG methods for computational electromagnetics
 J. Sci. Comput
"... Abstract. In this paper we introduce a new RKDG method for problems of wave propagation that achieves full highorder convergence in time and space. The novelty of the method resides in the way in which it marches in time. It uses an mthorder, mstage, low storage SSPRK scheme which is an extensio ..."
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Cited by 19 (1 self)
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Abstract. In this paper we introduce a new RKDG method for problems of wave propagation that achieves full highorder convergence in time and space. The novelty of the method resides in the way in which it marches in time. It uses an mthorder, mstage, low storage SSPRK scheme which is an extension to a class of nonautonomous linear systems of a recently designed method for autonomous linear systems. This extension allows for a highorder accurate treatment of the inhomogeneous, timedependent terms that enter the semidiscrete problem on account of the physical boundary conditions. Thus, if polynomials of degree k are used in the space discretization, the RKDG method is of overall order m = k + 1, for any k> 0. Moreover, we also show that the attainment of highorder spacetime accuracy allows for an efficient implementation of postprocessing techniques that can double the convergence order. We explore this issue in a onedimensional setting and show that the superconvergence of fluxes previously observed in full spacetime DG formulations is also attained in our new RKDG scheme. This allows for the construction of higherorder solutions via local interpolating polynomials. Indeed, if polynomials of degree k are used in the space discretization together with a timemarching method of order 2 k +1, a postprocessed approximation of order 2 k +1 is obtained. Numerical results in one and two space dimensions are presented that confirm the predicted convergence properties.