@TechReport{ it:2017-013,
author = {Davide Bianchi and Stefano Serra-Capizzano},
title = {Spectral Analysis of Finite-Difference Approximations of
$1-d$ Waves in Non-Uniform Grids},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2017},
number = {2017-013},
month = jul,
abstract = {Preserving the finite positive velocity of propagation of
continuous solutions of wave equations is one of the key
issues, when building numerical approximation schemes for
control and inverse problems. And this is hard to achieve
uniformly on all possible ranges of numerical solutions. In
particular, high frequencies often generate spurious
numerical solutions, behaving in a pathological manner and
making the propagation properties of continuous solutions
fail. The latter may lead to the divergence of the ``most
natural" approximation procedures for numerical control or
identification problems.
On the other hand, the velocity of propagation of high
frequency numerical wave-packets, the so-called group
velocity, is well known to be related to the spectral gap
of the corresponding discrete spectra. Furthermore most
numerical schemes in uniform meshes fail to preserve the
uniform gap property and, consequently, do not share the
propagation properties of continuous waves.
However, recently, S. Ervedoza, A. Marica and the E. Zuazua
have shown that, in $1-d$, uniform propagation properties
are ensured for finite-difference schemes in suitable
non-uniform meshes behaving in a monotonic manner. The
monotonicity of the mesh induces a preferred direction of
propagation for the numerical waves. In this way, meshes
that are suitably designed can ensure that all numerical
waves reach the boundary in an uniform time, which is the
key for the fulfillment of boundary controllability
properties.
In this paper we study the gap of discrete spectra of the
Laplace operator in $1-d$ for non-uniform meshes, analysing
the corresponding spectral symbol, which allows to show how
to design the discretization grid for improving the gap
behaviour. The main tool is the study of an univariate
monotonic version of the spectral symbol, obtained by
employing a proper rearrangement.
The analytical results are illustrated by a number of
numerical experiments. We conclude discussing some open
problems.}
}