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On the accurate long-time solution of the wave equation in exterior domains: asymptotic expansions and corrected boundary conditions

Authors: Thomas Hagstrom, S. I. Hariharan and R. C. MacCamy
Journal: Math. Comp. 63 (1994), 507-539
MSC: Primary 65N12; Secondary 35B40
MathSciNet review: 1248970
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Abstract: We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study both the short- and long-time behavior of the error. It is proved that, in two space dimensions, no local in time, constant-coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable-coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant-coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions, using energy methods and leading to asymptotically correct error bounds.

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Keywords: Boundary conditions, asymptotic expansions, wave equation
Article copyright: © Copyright 1994 American Mathematical Society