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Hyperbolicity of second-order in space systems of evolution equations

Hyperbolicity of second-order in space systems of evolution equations
Hyperbolicity of second-order in space systems of evolution equations
A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower-order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second-order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds.
0264-9381
S387-S404
Gundlach, Carsten
586f1eb5-3185-4b2b-8656-c29c436040fc
Martín-García, José M.
4d46af63-2651-477e-b0f0-67245eba67f0
Gundlach, Carsten
586f1eb5-3185-4b2b-8656-c29c436040fc
Martín-García, José M.
4d46af63-2651-477e-b0f0-67245eba67f0

Gundlach, Carsten and Martín-García, José M. (2006) Hyperbolicity of second-order in space systems of evolution equations. Classical and Quantum Gravity, 23 (16), S387-S404. (doi:10.1088/0264-9381/23/16/S06).

Record type: Article

Abstract

A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a reduction to first order and give necessary and sufficient criteria for strong/symmetric hyperbolicity of the reduction in terms of the principal part of the original second-order system. An alternative definition of strong hyperbolicity is based on the existence of a complete set of characteristic variables, and an alternative definition of symmetric hyperbolicity is based on the existence of a conserved (up to lower-order terms) energy. Both these definitions are made without any explicit reduction. Finally, strong hyperbolicity can be defined through a pseudo-differential reduction to first order. We prove that both definitions of symmetric hyperbolicity are equivalent and that all three definitions of strong hyperbolicity are equivalent (in three space dimensions). We show how to impose maximally dissipative boundary conditions on any symmetric hyperbolic second-order system. We prove that if the second-order system is strongly hyperbolic, any closed constraint evolution system associated with it is also strongly hyperbolic, and that the characteristic variables of the constraint system are derivatives of a subset of the characteristic variables of the main system, with the same speeds.

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Published date: 21 August 2006

Identifiers

Local EPrints ID: 48961
URI: https://eprints.soton.ac.uk/id/eprint/48961
ISSN: 0264-9381
PURE UUID: 47510e78-c189-4de1-ad36-ed6779a507fc
ORCID for Carsten Gundlach: ORCID iD orcid.org/0000-0001-9585-5375

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Date deposited: 18 Oct 2007
Last modified: 20 Jul 2018 00:34

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