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22
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
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...
A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
, 2008
"... ..."
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
Accurate radiation boundary conditions for the timedependent wave equation on unbounded domains
 I. J. for Numerical Methods in Engineering
, 1999
"... Asymptotic and exact local radiation boundary conditions (RBC) for the scalar timedependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local ..."
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Cited by 15 (8 self)
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Asymptotic and exact local radiation boundary conditions (RBC) for the scalar timedependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the nite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related nonreflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form highorder accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition.
Global discrete artificial boundary conditions for timedependent wave propagation
 J. Comput. Phys
"... We construct global artificial boundary conditions (ABCs) for the numerical simulation of wave processes on unbounded domains using a special nondeteriorating algorithm that has been developed previously for the longterm computation of waveradiation solutions. The ABCs are obtained directly for t ..."
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Cited by 11 (5 self)
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We construct global artificial boundary conditions (ABCs) for the numerical simulation of wave processes on unbounded domains using a special nondeteriorating algorithm that has been developed previously for the longterm computation of waveradiation solutions. The ABCs are obtained directly for the discrete formulation of the problem; in so doing, neither a rational approximation of “nonreflecting kernels” nor discretization of the continuous boundary conditions is required. The extent of temporal nonlocality of the new ABCs appears fixed and limited; in addition, the ABCs can handle artificial boundaries of irregular shape on regular grids with no fitting/adaptation needed and no accuracy loss induced. The nondeteriorating algorithm, which is the core of the new ABCs, is inherently threedimensional, it guarantees temporally uniform grid convergence of the solution driven by a continuously operating source on arbitrarily long time intervals and provides unimprovable linear computational complexity with respect to the grid dimension. The algorithm is based on the presence of lacunae, i.e., aft fronts of the waves, in wavetype solutions in odddimensional spaces. It can, in fact, be built as a modification on top of any
Radiation Boundary Conditions for Maxwell's Equations: A Review of Accurate Time{domain Formulations
 J. Comp. Math
"... We review timedomain formulations of radiation boundary conditions for Maxwell’s equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on iden ..."
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Cited by 11 (2 self)
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We review timedomain formulations of radiation boundary conditions for Maxwell’s equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.
Computation of Far Field Solutions Based on Exact Nonreflecting Boundary Conditions for the TimeDependent Wave Equation
, 1999
"... In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (farfield) solution for timedependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a ..."
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Cited by 6 (4 self)
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In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (farfield) solution for timedependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the nearfield solution, are imposed as Cauchy data for the radial wave equation in the farfield. Similar to the farfield computation scheme used by Wright, the radial modes in the exterior region are computed using an explicit finite difference solver. However, instead of using an `infinite grid', we truncate the exterior radial grid at the farfield point of interest, and for each harmonic, impose the same exact NRBC used for the nearfield truncation boundary, here expressed in modal form. Using this approach, two different methods for extrapolating the nearfield solution to the farfield are possible. In the first, the near fie...