Hyperbolic extensions of constrained PDEs
Systems of partial differential equations (PDEs) comprising 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 Cauch...
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Frontiers Media S.A.
2025-02-01
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| Series: | Frontiers in Physics |
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| Online Access: | https://www.frontiersin.org/articles/10.3389/fphy.2024.1517192/full |
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| author | Fernando Abalos Oscar Reula David Hilditch |
| author_facet | Fernando Abalos Oscar Reula David Hilditch |
| author_sort | Fernando Abalos |
| collection | DOAJ |
| description | Systems of partial differential equations (PDEs) comprising 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. In this article, we first review the use of hyperbolic reductions, where the evolution equations are singled out for consideration. We then examine in greater detail the extensions, namely, systems in which constraints are evolved as auxiliary variables alongside the original variables, resulting in evolution systems with no constraints. Assuming a particular structure of the original system, we provide sufficient conditions for the strong hyperbolicity of an extension. Finally, this theory is applied to the examples of electromagnetism and a toy model of magnetohydrodynamics. |
| format | Article |
| id | doaj-art-2db87a476a3440da973ae8a3d6d8faa9 |
| institution | OA Journals |
| issn | 2296-424X |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Physics |
| spelling | doaj-art-2db87a476a3440da973ae8a3d6d8faa92025-08-20T02:13:01ZengFrontiers Media S.A.Frontiers in Physics2296-424X2025-02-011210.3389/fphy.2024.15171921517192Hyperbolic extensions of constrained PDEsFernando Abalos0Oscar Reula1David Hilditch2Departament de Física and Institute of Applied Computing & Community Code (IAC3), Universitat de les Illes Balears, Palma deMallorca, SpainFacultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba and IFEG-CONICET, Ciudad Universitaria, Córdoba, ArgentinaCENTRA, Departamento de Física, Instituto Superior Técnico IST, Universidade de Lisboa UL, Lisboa, PortugalSystems of partial differential equations (PDEs) comprising 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. In this article, we first review the use of hyperbolic reductions, where the evolution equations are singled out for consideration. We then examine in greater detail the extensions, namely, systems in which constraints are evolved as auxiliary variables alongside the original variables, resulting in evolution systems with no constraints. Assuming a particular structure of the original system, we provide sufficient conditions for the strong hyperbolicity of an extension. Finally, this theory is applied to the examples of electromagnetism and a toy model of magnetohydrodynamics.https://www.frontiersin.org/articles/10.3389/fphy.2024.1517192/fullwell-posed initial value problemconstraint equationsevolution equationsextensionssingular value decomposition (SVD)Kronecker decomposition |
| spellingShingle | Fernando Abalos Oscar Reula David Hilditch Hyperbolic extensions of constrained PDEs Frontiers in Physics well-posed initial value problem constraint equations evolution equations extensions singular value decomposition (SVD) Kronecker decomposition |
| title | Hyperbolic extensions of constrained PDEs |
| title_full | Hyperbolic extensions of constrained PDEs |
| title_fullStr | Hyperbolic extensions of constrained PDEs |
| title_full_unstemmed | Hyperbolic extensions of constrained PDEs |
| title_short | Hyperbolic extensions of constrained PDEs |
| title_sort | hyperbolic extensions of constrained pdes |
| topic | well-posed initial value problem constraint equations evolution equations extensions singular value decomposition (SVD) Kronecker decomposition |
| url | https://www.frontiersin.org/articles/10.3389/fphy.2024.1517192/full |
| work_keys_str_mv | AT fernandoabalos hyperbolicextensionsofconstrainedpdes AT oscarreula hyperbolicextensionsofconstrainedpdes AT davidhilditch hyperbolicextensionsofconstrainedpdes |