@PhDThesis{ itlic:2010-004,
author = {Kenneth Duru},
title = {Perfectly Matched Layers for Second Order Wave Equations},
school = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2010},
number = {2010-004},
type = {Licentiate thesis},
month = may,
day = {7},
pages = {102},
copies-printed= {80},
abstract = {Numerical simulation of propagating waves in unbounded
spatial domains is a challenge common to many branches of
engineering and applied mathematics. Perfectly matched
layers (PML) are a novel technique for simulating the
absorption of waves in open domains. The equations modeling
the dynamics of phenomena of interest are usually posed as
differential equations (or integral equations) which must
be solved at every time instant. In many application areas
like general relativity, seismology and acoustics, the
underlying equations are systems of second order hyperbolic
partial differential equations. In numerical treatment of
such problems, the equations are often rewritten as first
order systems and are solved in this form. For this reason,
many existing PML models have been developed for first
order systems. In several studies, it has been reported
that there are drawbacks with rewriting second order
systems into first order systems before numerical solutions
are obtained. While the theory and numerical methods for
first order systems are well developed, numerical
techniques to solve second order hyperbolic systems is an
on-going research.
In the first part of this thesis, we construct PML
equations for systems of second order hyperbolic partial
differential equations in two space dimensions, focusing on
the equations of linear elasto-dynamics. One advantage of
this approach is that we can choose auxiliary variables
such that the PML is strongly hyperbolic, thus strongly
well-posed. The second is that it requires less auxiliary
variables as compared to existing first order formulations.
However, in continuum the stability of both first order and
second order formulations are linearly equivalent. A
turning point is in numerical approximations. We have found
that if the so-called geometric stability condition is
violated, approximating the first order PML with standard
central differences leads to a high frequency instability
for any given resolution. The second order discretization
behaves much more stably. In the second order setting
instability occurs only if unstable modes are well resolved.
The second part of this thesis discusses the construction
of PML equations for the time-dependent Schr\"odinger
equation. From mathematical perspective, the Schr\"odinger
equation is unique, in the sense that it is only first
order in time but second order in space. However, with
slight modifications, we carry over our ideas from the
hyperbolic systems to the Schr\"odinger equations and
derive a set of asymptotically stable PML equations. The
new model can be viewed as a modified complex absorbing
potential (CAP). The PML model can easily be adapted to
existing codes developed for CAP by accurately discretizing
the auxiliary variables and appending them accordingly.
Numerical experiments are presented illustrating the
accuracy and absorption properties of the new PML model.
We are hopeful that the results obtained in this thesis
will find useful applications in time-dependent wave
scattering calculations.}
}