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BRIEF RESEARCH REPORT article

Front. Phys.
Sec. Interdisciplinary Physics
Volume 12 - 2024 | doi: 10.3389/fphy.2024.1517192
This article is part of the Research Topic Quasi-Normal Modes, Non-Selfadjoint Operators and Pseudospectrum: an Interdisciplinary Approach View all 12 articles

Hyperbolic extensions of constrained PDEs

Provisionally accepted
Fernando Abalos Fernando Abalos 1*Oscar Reula Oscar Reula 2David Hilditch David Hilditch 3
  • 1 University of the Balearic Islands, Palma de Mallorca, Spain
  • 2 National University of Cordoba, Córdoba, Córdoba, Argentina
  • 3 Associação do Instituto Superior Técnico de Investigação e Desenvolvimento (IST-ID), Lisboa, Portugal

The final, formatted version of the article will be published soon.

    Systems of PDEs comprised of a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of the well-posedness of the Cauchy problem for these systems. Presently we review the use of hyperbolic reductions, in which the evolution equations are singled out for consideration. We then examine in greater detail the extensions, in which constraints are evolved as auxiliary variables alongside the original variables. Assuming a particular structure of the original system, we give sufficient conditions for strong-hyperbolicity of an extension. This theory is then applied to the examples of electromagnetism and a toy for magnetohydrodynamics.

    Keywords: Well posed initial value problem, Constraint equations, Evolution equations, Extensions, singular value decomposition, Kronecker decomposition, electromagnetism, magnetohydrodynamics

    Received: 25 Oct 2024; Accepted: 30 Dec 2024.

    Copyright: © 2024 Abalos, Reula and Hilditch. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

    * Correspondence: Fernando Abalos, University of the Balearic Islands, Palma de Mallorca, Spain

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