Hyperbolic systems with relaxation: characterization of stiff well-posedness and asymptotic expansions
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1224322
- author
- Schroll, Achim LU and Lorenz, Jens
- publishing date
- 1999
- 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
- http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WK2-45FKS31-3T-1&_cdi=6894&_user=745831&_orig=search&_coverDate=07%2F15%2F1999&_sk=997649997&view=c&wchp=dGLbVzW-zSkzV&md5=1d528ede24fffa63eb89597bec726dc6&ie=/sdarticle.pdf
- 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 < 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}}, }