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Inhomogeneous Conditions at Open Boundaries for Wave Propagation Problems,
STANFORD UNIV CA CENTER FOR LARGE SCALE SCIENTIFIC COMPUTATION
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When wave propagation problems are solved numerically, it is often necessary to introduce artificial boundaries. In most cases no data are available at these boundaries, and therefore it is necessary to construct boundary conditions which in some way accounts for the behavior of the solution outside the computational domain. Hence some assumption must be made which makes it possible to solve the problem exactly outside, or alternatively to compute the solution approximately in a simple way. One such assumption is that the waves are propagating only in the outward direction across the boundary. This is the basis for various procedures in common use, the most general class being the absorbing boundary conditions. The absorbing boundary conditions of higher order are formulated in terms of differentiated functions. This leads necessarily to weakly ill-posed problems, and as a consequence also to unstable numerical methods. This has been pointed out for the scalar wave-equation. For conditions of order two or less, the possible effects of the instability is outweighed by the small reflection coefficients. This paper we investigate this problem further. When the initial data do not have compact support within the computational domain, or when some outside source is present, the boundary conditions become inhomogeneous. This tends to amplify the influence of the instability, and it becomes essential to modify the boundary procedure.
APPROVED FOR PUBLIC RELEASE