Results 1  10
of
24
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 ..."
Abstract

Cited by 219 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 126 (19 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 91 (3 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 55 (2 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
(Show Context)
. 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 ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
(Show Context)
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 ..."
(Show Context)
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 ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 15 (8 self)
 Add to MetaCart
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.
Characteristic Evolution and Matching
, 2008
"... I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativ ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
(Show Context)
I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to eliminate the role of this artificial outer boundary via Cauchycharacteristic matching, by which the radiated waveform can be computed at null infinity. Progress in this direction is discussed.