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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
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)
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date added to LUP
2008-09-02 09:13:58
date last changed
2017-01-01 07:47:40
@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},
  keyword      = {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},
  volume       = {235},
  year         = {1999},
}