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147
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
On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer
 J. Comput. Phys
, 1995
"... waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative ..."
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Cited by 59 (1 self)
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waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane wave solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing outgoing waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.
A New Discrete Transparent Boundary Condition for Standard and Wide Angle "Parabolic" Equations in Underwater Acoustics
"... This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nic ..."
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Cited by 48 (16 self)
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This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC is derived from the discrete whole–space problem that yields an unconditionally stable scheme. The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems.
Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 46 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
, 2008
"... ..."
Fictitious Domain Methods For The Numerical Solution Of ThreeDimensional Acoustic Scattering Problems
 J. Comput. Phys
, 1999
"... . Efficient iterative methods for the numerical solution of threedimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. ..."
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Cited by 29 (17 self)
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. Efficient iterative methods for the numerical solution of threedimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. The finite element discretization of the approximate boundary value problem is performed using locally fitted meshes, and algebraic fictitious domain methods with separable preconditioners are applied to the solution of the arising mesh equations. These methods are based on imbedding the original domain into a larger one with a simple geometry (for example, a sphere or a parallelepiped). The iterative solution method is realized in a lowdimensional subspace, and partial solution methods are applied to the linear systems with the preconditioner. Results of numerical experiments demonstrate the efficiency and accuracy of the approach. Key words. Acoustic scattering, nonreflecting boundary ...
Highorder nonreflecting boundary scheme for timedependent waves
 Journal of Computational Physics
"... waves ..."
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