Hyperbolic extensions of constrained PDEs
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. Pr...
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| Main Authors | , , |
|---|---|
| Format | Journal Article |
| Language | English |
| Published |
23.10.2024
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| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Abalos, Fernando Reula, Oscar Hilditch, David |
| Author_xml | – sequence: 1 givenname: Fernando surname: Abalos fullname: Abalos, Fernando – sequence: 2 givenname: Oscar surname: Reula fullname: Reula, Oscar – sequence: 3 givenname: David surname: Hilditch fullname: Hilditch, David |
| BackLink | https://doi.org/10.48550/arXiv.2410.18286$$DView paper in arXiv |
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| Snippet | Systems of PDEs comprised of a combination of constraints and evolution
equations are ubiquitous in physics. For both theoretical and practical
reasons, such... |
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| SubjectTerms | Mathematics - Analysis of PDEs Physics - General Relativity and Quantum Cosmology |
| Title | Hyperbolic extensions of constrained PDEs |
| URI | https://arxiv.org/abs/2410.18286 |
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