Mathematical Sciences: Non-Reflecting Boundary Conditions Based on Far Field Expansions

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9530937 Bayliss The effectiveness of non-reflecting boundary conditions for wave propagation problems which are based on far field expansions of the solution will be developed, implemented and assessed. Primarily transient wave propagation problems will be considered, although many of the techniques will also be applicable in the frequency domain. The boundary conditions are based on first deriving convergent or asymptotic expansions of the solution valid in the far field and then constructing differential operators which annihilate the leading order terms in the expansions. Generally these operators require specification of an origin from which outgoing waves are assumed to emanate. Techniques will be developed in which interior information is used to improve the performance of the boundary condition. In particular, adaptive boundary conditions will be developed in which the origin is estimated dynamically and locally from properties of the solution in the interior. Techniques to increase the order of the boundary condition without increasing the order of the differential operator will also be developed. This will be done by incorporating inhomogeneous terms into the boundary operator, again using properties of the computed solution in the interior. Finally, far field expansions and associated boundary conditions suitable for damping layers will be developed, thereby allowing a reduction in size of the layers. The numerical computation of waves, for example, acoustic and electromagnetic waves, will be considered. These problems are typically governed by equations specified in regions which are unbounded in at least one direction. In order for the problem to be formulated on a computer it is necessary that the unbounded region be replaced by a bounded region. Furthermore, it is necessary that some boundary condition be imposed at the edge (boundary) of the bounded region. This boundary condition should ideally simulate the ori ginal unbounded problem. However, in general the boundary treatment will not be a perfect simulation of the unbounded problem and there will be spurious reflections emanating from the boundary. These reflections will propagate into the interior region and seriously degrade the accuracy of the computed solution. Thus, it is essential for accurate computation that these spurious reflections be minimized. Boundary conditions will be considered based on properties of the waves in the far field, i.e., far from the sources where the waves are generated. Currently for any given computation the boundary conditions are generally determined in advance, i.e., at the beginning of the computation, and are not changed as the solution evolves in time. Thus, the boundary conditions make no use of information about the waves in the interior. Methodologies will be developed in which interior information can be used to improve the performance of the boundary conditions. In this approach, the boundary conditions will change as time evolves, adapting to the nature of the solution by making use of interior information. It is anticipated that the resulting boundary treatment will allow a significant reduction in spurious reflections. Thus, the proposed new boundary conditions offer the prospect of significant improvement in efficiency and accuracy for the computation of wave propagation problems. Applications to acoustic, electromagnetic and elastic wave propagation will be considered.