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16
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
Rapid Evaluation of Radiation Boundary Kernels for Timedomain Wave Propagation on Blackholes
, 2004
"... For scalar, electromagnetic, or gravitational wave propagation on a background Schwarzschild blackhole, we describe the exact nonlocal radiation outer boundary conditions (robc) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the robc is based on La ..."
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Cited by 17 (3 self)
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For scalar, electromagnetic, or gravitational wave propagation on a background Schwarzschild blackhole, we describe the exact nonlocal radiation outer boundary conditions (robc) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the robc is based on Laplace and spherical– harmonic transformation of the Regge–Wheeler equation, the pde governing the wave propagation, with the resulting radial ode an incarnation of the confluent Heun equation. For a given angular index l the robc feature integral convolution between a time–domain radiation boundary kernel (tdrk) and each of the corresponding 2l + 1 spherical–harmonic modes of the radiating wave. The tdrk is the inverse Laplace transform of a frequency–domain radiation kernel (fdrk) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ode. We numerically implement the robc via a rapid algorithm involving approximation of the fdrk by a rational function. Such an approximation is tailored to have relative error ε uniformly along the axis of imaginary Laplace frequency. Theoretically, ε is also a long–time bound on the relative convolution error. Via study of one–dimensional radial evolutions, we demonstrate that the robc capture the phenomena of quasinormal ringing and decay tails. Moreover, carrying out a numerical experiment in which a wave packet strikes the boundary at an angle, we find that the robc yield accurate results in a three–dimensional setting. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (agh), save for one key difference. Whereas agh had the usual armamentarium of analytical results (asymptotics, order recursion relations, bispectrality) for Bessel functions at their disposal, what we need to know about Heun functions must be gathered numerically as relatively less is known about them. Therefore, unlike agh, we are unable to offer an asymptotic analysis of our rapid implementation.
The numerical calculations of traveling wave solutions of nonlinear parabolic equations
"... Abstract. Traveling wave solutions have been studied for a variety of nonlinear parabolic problems. In the initial value approach to such problems the initial data at infinity determines the wave that propagates. The numerical simulation of such problems is thus quite difficult. If the domain is rep ..."
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Cited by 7 (0 self)
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Abstract. Traveling wave solutions have been studied for a variety of nonlinear parabolic problems. In the initial value approach to such problems the initial data at infinity determines the wave that propagates. The numerical simulation of such problems is thus quite difficult. If the domain is replaced by a finite one, to facilitate numerical computations, then appropriate boundary conditions on the "artificial " boundaries must depend upon the initial data in the discarded region. In this work we derive such boundary conditions, based on the Laplace transform of the linearized problems at +co, and illustrate their utility by presenting a numerical solution of Fisher’s equation which has been proposed as a model in genetics. Key words, artificial boundary conditions, parabolic traveling waves AMS(MOS) subject classifications. 65M99, 35K15 1. Introduction. We
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...
Large Eddy Simulation in a Channel with Exit Boundary Conditions
 ADVANCES IN MULTIFLUID FLOWS. PROCEEDINGS OF THE AMSIMSSIAM JOINT SUMMER RESEARCH CONFERENCE ON MULTIFLUID FLOWS AND INTERFACIAL
, 1996
"... The influence of the exit boundary conditions (vanishing first derivative of the velocity components and constant pressure) on the large eddy simulation of the fully developed turbulent channel flow has been investigated for equidistant and stretched grids at the channel exit. Results show that the ..."
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Cited by 5 (0 self)
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The influence of the exit boundary conditions (vanishing first derivative of the velocity components and constant pressure) on the large eddy simulation of the fully developed turbulent channel flow has been investigated for equidistant and stretched grids at the channel exit. Results show that the chosen exit boundary conditions introduce some small disturbance which is mostly damped by the grid stretching. The difference between the fully developed turbulent channel flow obtained with LES with periodicity condition and the inlet and exit and the LES with fully developed flow at the inlet and the exit boundary condition is less than 10 % for equidistant grids and less than 5 % for the case grid stretching. The chosen boundary condition is of interest because it may be used in complex flows with backflow at exit.
A FarField Nonreflecting Boundary Condition for TwoDimensional Wake Flows
, 1995
"... Farfield boundary conditions for external flow problems have been developed based upon longwave perturbations of linearized flow equations about a steady state far field solution. The boundary improves convergence to steady state in singlegrid temporal integration schemes using both regulartime ..."
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Cited by 2 (0 self)
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Farfield boundary conditions for external flow problems have been developed based upon longwave perturbations of linearized flow equations about a steady state far field solution. The boundary improves convergence to steady state in singlegrid temporal integration schemes using both regulartimestepping and localtimestepping. The farfield boundary may be near the trailing edge of the body which significantly reduces the number of grid points, and therefore the computational time, in the numerical calculation. In addition the solution produced is smoother in the farfield than when using extrapolation conditions. The boundary condition maintains the convergence rate to steady state in schemes utilizing multigrid acceleration.
Gustafsson� Inhomogeneous Conditions at Open Boundaries for Wave Propagation Problems
 Appl. Numer. Math
, 1988
"... 'ECO ol.'.. omoo ..."
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