Boundary waves and stability of the perfectly matched layer II: extensions to first order systems and numerical stability
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Authors
Duru, Kenneth
Kozdon, Jeremy E.
Kreiss, Gunilla
Subjects
elastic wave equation
rst order systems
Rayleigh surface waves
perfectly matched layers
stability
normal mode analysis
high order nite di erence
summation{by{parts
penalty method
rst order systems
Rayleigh surface waves
perfectly matched layers
stability
normal mode analysis
high order nite di erence
summation{by{parts
penalty method
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2012
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Abstract
In this paper we study the stability of the perfectly matched layer (PML) for the elastic wave equation in rst order form.
The theory of temporal stability of initial value problems corresponding to the PML is well developed. For initial boundary value problems
(IBVPs) the theory of temporal stability of the PML is less complete. First, we study the solutions of two IBVPs corresponding to the PML
for the elastic wave equation in rst order form. We consider separately a PML on the lower half-plane with free-surface boundary conditions
at y = 0 and a PML on the left half-plane with characteristic boundary conditions at x = . In both cases the PML truncates a boundary
in the x{direction. Using normal mode analysis we prove that the lower half{plane problem and the left half{plane problem do not support
temporally growing modes.
Second, we develop a high order accurate nite di erence approximation of the PML subject to the boundary conditions. To enable accurate
and stable boundary treatments for the PML we construct continuous energy estimates in the Laplace space. We use summation-by-parts
nite di erence operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. By mimicking the
continuous energy estimates in the discrete setting, we construct stable numerical boundary procedures for the PML subject to the free-surface
and the characteristic boundary conditions. Numerical experiments are presented corroborating the theoretical results.
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Applied Mathematics
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.