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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.18286 |
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| _version_ | 1866909361750671360 |
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| author | Abalos, Fernando Reula, Oscar Hilditch, David |
| author_facet | Abalos, Fernando Reula, Oscar Hilditch, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolic extensions of constrained PDEs Abalos, Fernando Reula, Oscar Hilditch, David Analysis of PDEs General Relativity and Quantum Cosmology 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. |
| title | Hyperbolic extensions of constrained PDEs |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2410.18286 |