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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.
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
Towards accurate artificial boundary conditions for nonlinear PDEs through examples
, 2009
"... The aim of this paper is to give a comprehensive review of current developments related to the derivation of artificial boundary conditions for nonlinear partial differential equations. The essential tools to build such boundary conditions are based on pseudodifferential and paradifferential calcu ..."
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Cited by 7 (1 self)
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The aim of this paper is to give a comprehensive review of current developments related to the derivation of artificial boundary conditions for nonlinear partial differential equations. The essential tools to build such boundary conditions are based on pseudodifferential and paradifferential calculus. We present various derivations and compare them. Some numerical results illustrate their respective accuracy and analyze the potential of each technique.