Technical Report 2017-013

Spectral Analysis of Finite-Difference Approximations of 1-d Waves in Non-Uniform Grids

Davide Bianchi and Stefano Serra-Capizzano

July 2017

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.

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