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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...
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 ..."
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Cited by 12 (1 self)
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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.
An Application of the Difference Potentials Method to Solving External Problems in CFD
 Problems in CFD, NASA Technical Memorandum No. 110338, Langley Research Center
, 1997
"... Numerical solution of infinitedomain boundaryvalue problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the origi ..."
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Cited by 7 (2 self)
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Numerical solution of infinitedomain boundaryvalue problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinitedomain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available. One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABCs) at the newly formed external boundary. The role of the ABCs is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinitedoma...
An Improved Treatment of External Boundary for ThreeDimensional Flow Computations
 AIAA Paper
, 1997
"... We present an innovative numerical approach for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating threedimensional compressible viscous flows over finite bodies. The approach is based on application of the difference potentials method by ..."
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Cited by 4 (2 self)
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We present an innovative numerical approach for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating threedimensional compressible viscous flows over finite bodies. The approach is based on application of the difference potentials method by V. S. Ryaben'kii and extends our previous technique developed for the twodimensional case. The new boundary conditions methodology has been successfully combined with the NASAdeveloped code TLNS3D and used for the analysis of wingshaped configurations in subsonic and transonic flow regimes. As demonstrated by the computational experiments, the improved external boundary conditions allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of convergence of the multigrid iterations.
JOURNAL OF COMPUTATIONAL PHYSICS 136, 140–167 (1997) ARTICLE NO. CP975754 Cauchy–Characteristic Evolution and Waveforms
, 1996
"... These are the conditions responsible for the proper 1/r asymptotic decay of the radiation fields. However, for pracWe investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global tical purposes, in the computational treatment of such a ..."
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These are the conditions responsible for the proper 1/r asymptotic decay of the radiation fields. However, for pracWe investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global tical purposes, in the computational treatment of such a spacetime evolution algorithm based on a Cauchy initialvalue for system by a Cauchy initialvalue problem, an outer boundmulation in a bounded interior region and based on characteristic ary is artificially established at some large but finite dishypersurfaces in the exterior; we match the two schemes at their tance lying in the wave zone, i.e., many wavelengths fromcommon interface. The characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate the source. Some outgoing radiation condition is then iminterval, so that our numerical solution extends to infinity and accu posed upon this boundary in an attempt to approximate rately models the freespace problem. The matching interface need the proper asymptotic behavior at infinity. The boundary not be situated far from the sources, the wavefronts may have condition may cause partial reflection of the outgoing wavearbitrary nonspherical geometry, and strong nonlinearity may be back into the system [1–4], which contaminates the accupresent in both the interior and the exterior regions. Stability and secondorder convergence of the algorithms (to the exact solution racy of the evolution and of the waveform determined of the infinitedomain problem) are established numerically in three at infinity. Furthermore, nonlinear wave equations often