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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.
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...
Discretely nonreflecting boundary conditions for linear hyperbolic systems
 J. Comput. Phys
, 2000
"... Many compressible flow and aeroacoustic computations rely on accurate nonreflecting or radiation boundary conditions. When the equations and boundary conditions are discretized using a finitedifference scheme, the dispersive nature of the discretized equations can lead to spurious numerical refle ..."
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Cited by 14 (2 self)
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Many compressible flow and aeroacoustic computations rely on accurate nonreflecting or radiation boundary conditions. When the equations and boundary conditions are discretized using a finitedifference scheme, the dispersive nature of the discretized equations can lead to spurious numerical reflections not seen in the continuous boundary value problem. Here we construct discretely nonreflecting boundary conditions, which account for the particular finitedifference scheme used, and are designed to minimize these spurious numerical reflections. Stable boundary conditions that are local and nonreflecting to arbitrarily high order of accuracy are obtained, and test cases are presented for the linearized Euler equations. For the cases presented, reflections for a pressure pulse leaving the boundary are reduced by up to two orders of magnitude over typical ad hoc closures, and for a vorticity pulse, reflections are reduced by up to four orders of magnitude. c ° 2000 Academic Press Key Words: nonreflecting boundary conditions; artificial boundary conditions; finite difference; Euler equations; highorderaccurate methods. CONTENTS
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions
, 2005
"... In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourthorder numerical method for solving the nonlinear Helmholtz equation which governs the propagation of timeharmonic laser beams in media with a Kerrtype nonlinearity. A key element of the algorithm was a new nonlocal twoway artificial b ..."
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Cited by 7 (1 self)
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In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourthorder numerical method for solving the nonlinear Helmholtz equation which governs the propagation of timeharmonic laser beams in media with a Kerrtype nonlinearity. A key element of the algorithm was a new nonlocal twoway artificial boundary condition (ABC), set in the direction of beam propagation. This twoway ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632–677] has allowed for a direct simulation of nonlinear selffocusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeldtype local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a nonHermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical
Low Mach Number Asymptotics of the NavierStokes Equations and Numerical Implications
, 1999
"... Low Mach number asymptotics of the NavierStokes equations reveals the role of the large global thermodynamic pressure, the small acoustic pressure and the very small 'incompressible' pressure. Solving for the changes of the conservative variables with respect to stagnation conditions reta ..."
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Cited by 7 (0 self)
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Low Mach number asymptotics of the NavierStokes equations reveals the role of the large global thermodynamic pressure, the small acoustic pressure and the very small 'incompressible' pressure. Solving for the changes of the conservative variables with respect to stagnation conditions retains the conservative discretization and avoids the cancellation problem, when computing the small changes in low Mach number flow. Key words: NavierStokes equations, asymptotic analysis, low Mach number flow, aeroacoustics, cancellation, perturbation formulation, finite volume method, approximate Riemann solver. Notation: In this lecture, all physical quantities with superscript * are dimensional and all physical quantities without superscript * are nondimensional. Unless stated otherwise, the nondimensionalization (13) is used.
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 ..."
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Cited by 6 (3 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 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...
2001: Towards a transparent boundary condition for compressible NavierStokes equations
 Int. J. Numer. Meth. Fluids
"... A new artificial boundary condition for 2D subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow as well as a convection velocity are used ..."
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Cited by 5 (0 self)
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A new artificial boundary condition for 2D subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow as well as a convection velocity are used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems whose solution is known and comparisons with other boundary conditions show the efficiency of the boundary condition. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow.