Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach
type: Item
instance of: scholarly article
title: Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach
zbMATH Open document ID: 1375.81163
author: Aurora Marica
author: Enrique Zuazua
published in: Foundations of Computational Mathematics
publication date: 2015-12-04T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11824/237
review text: The article focusses on an analysis of the propagation properties of discrete waves while solving the finite difference schemes for two 1D wave propagation models, namely the first-order transport equation and second-order wave equation. The fundamental contribution of this work is the provision of a structured meaning to notions such as the \textit{principal symbol} of the discrete wave operator, allowing one to construct bi-characteristic and characteristic rays propagating the energy of solutions. Boundary controllability and identifiability properties of the solutions of the wave equation hold due to the fact that the energy of solutions is driven by characteristics that reach the boundary where controllers or observers are placed. This property is known to fail in general for numerical schemes due to high-frequency spurious numerical solutions. However this analysis has thus far only been done within the context of uniform meshes. This research extends this by considering non-uniform meshes and developing the tools and notions required for microlocal analysis.
reviewed by: Charis Harley
Mathematics Subject Classification ID: 81S30
Mathematics Subject Classification ID: 35A21
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 70K05
Mathematics Subject Classification ID: 81T80
zbMATH DE Number: 6516355
zbMATH Keywords: variable coefficients transport/wave equations
zbMATH Keywords: finite difference approximations
zbMATH Keywords: non-uniform meshes
zbMATH Keywords: Wigner transform/measure
zbMATH Keywords: bi-characteristic rays
zbMATH Keywords: umklapp/internal reflection
MaRDI profile type: Publication
OpenAlex ID: W1982368468
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DOI: 10.1007/S10208-014-9232-X
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