Results 1  10
of
13
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...
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...
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
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 ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
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.
Artificial Boundary Conditions Based On The Difference Potentials Method
 IN PROCEEDINGS OF THE SIXTH INTERNATIONAL SYMPOSIUM ON COMPUTATIONAL FLUID DYNAMICS, IV
, 1996
"... 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 to be most significant i ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
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 to be 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 present a wide class of practically important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms. 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 (computationally) expensiv...
Artificial boundary conditions for the numerical simulation of unsteady acoustic waves
 Author information Frédéric Nataf, Laboratory J.L. Lions, UPMC and CNRS
"... A central characteristic feature of an important class of hyperbolic PDEs in odddimension spaces is the presence of lacunae, i.e., sharp aft fronts of the waves, in their solutions. This feature, which is often associated with the Huygens ’ principle, is employed to construct accurate nonlocal arti ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
A central characteristic feature of an important class of hyperbolic PDEs in odddimension spaces is the presence of lacunae, i.e., sharp aft fronts of the waves, in their solutions. This feature, which is often associated with the Huygens ’ principle, is employed to construct accurate nonlocal artificial boundary conditions (ABCs) for the Maxwell equations. The setup includes the propagation of electromagnetic waves from moving sources over unbounded domains. For the purpose of obtaining a finite numerical approximation the domain is truncated, and the ABCs guarantee that the outer boundary be completely transparent for all the outgoing waves. The lacunaebased approach has earlier been used for the scalar wave equation, as well as for acoustics. In the current paper, we emphasize the key distinctions between those previously studied models and the Maxwell equations of electrodynamics, as they manifest themselves in the context of lacunaebased integration. The extent of temporal nonlocality of the proposed ABCs is fixed and limited, and this is not a result of any approximation, it is rather an immediate implication of the existence of lacunae. The ABCs can be applied to any numerical scheme that is used to integrate the Maxwell equations. They do not require any geometric adaptation, and they guarantee that the accuracy of the boundary treatment will not deteriorate even on long time intervals. The paper presents a number of numerical demonstrations that corroborate the theoretical design features of the boundary conditions.