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Hyperbolic systems with relaxation: characterization of stiff well-posedness and asymptotic expansions

Schroll, Achim LU and Lorenz, Jens (1999) In Journal of Mathematical Analysis and Applications 235(2). p.497-532
Abstract
The Cauchy problem for linear constant-coefficient hyperbolic systems ut + ∑j A(j)uxj = (1/δ)Bu + Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0 < δ 1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff... (More)
The Cauchy problem for linear constant-coefficient hyperbolic systems ut + ∑j A(j)uxj = (1/δ)Bu + Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0 < δ 1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness—which is a weaker requirement than the existence of an entropy—leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase flow in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem. (Less)
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author
and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
hyperbolic system, relaxation, well-posedness, asymptotic expansion, equilibrium limit, speed condition
in
Journal of Mathematical Analysis and Applications
volume
235
issue
2
pages
497 - 532
publisher
Elsevier
external identifiers
  • scopus:0346515637
ISSN
0022-247X
DOI
10.1006/jmaa.1999.6400
language
English
LU publication?
no
id
993f0483-c565-4b99-a1fc-11dd795b1c23 (old id 1224322)
alternative location
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date added to LUP
2016-04-04 09:22:25
date last changed
2022-01-29 17:33:04
@article{993f0483-c565-4b99-a1fc-11dd795b1c23,
  abstract     = {{The Cauchy problem for linear constant-coefficient hyperbolic systems ut + ∑j A(j)uxj = (1/δ)Bu + Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0 &lt; δ 1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness—which is a weaker requirement than the existence of an entropy—leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase flow in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem.}},
  author       = {{Schroll, Achim and Lorenz, Jens}},
  issn         = {{0022-247X}},
  keywords     = {{hyperbolic system; relaxation; well-posedness; asymptotic expansion; equilibrium limit; speed condition}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{497--532}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Mathematical Analysis and Applications}},
  title        = {{Hyperbolic systems with relaxation: characterization of stiff well-posedness and asymptotic expansions}},
  url          = {{http://dx.doi.org/10.1006/jmaa.1999.6400}},
  doi          = {{10.1006/jmaa.1999.6400}},
  volume       = {{235}},
  year         = {{1999}},
}