Hyperbolic systems with relaxation: characterization of stiff wellposedness and asymptotic expansions
(1999) In Journal of Mathematical Analysis and Applications 235(2). p.497532 Abstract
 The Cauchy problem for linear constantcoefficient 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 nontrivial nullspace. A concept of stiff wellposedness is introduced that ensures solution estimates independent of 0 < δ 1. Stiff wellposedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the socalled 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 constantcoefficient 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 nontrivial nullspace. A concept of stiff wellposedness is introduced that ensures solution estimates independent of 0 < δ 1. Stiff wellposedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the socalled equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff wellposedness—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 twophase flow in a porous medium and show stiff wellposedness 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:
http://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, wellposedness, 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
 0022247X
 DOI
 10.1006/jmaa.1999.6400
 language
 English
 LU publication?
 no
 id
 993f0483c5654b99a1fc11dd795b1c23 (old id 1224322)
 alternative location
 http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WK245FKS313T1&_cdi=6894&_user=745831&_orig=search&_coverDate=07%2F15%2F1999&_sk=997649997&view=c&wchp=dGLbVzWzSkzV&md5=1d528ede24fffa63eb89597bec726dc6&ie=/sdarticle.pdf
 date added to LUP
 20080902 09:13:58
 date last changed
 20180529 11:34:44
@article{993f0483c5654b99a1fc11dd795b1c23, abstract = {The Cauchy problem for linear constantcoefficient 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 nontrivial nullspace. A concept of stiff wellposedness is introduced that ensures solution estimates independent of 0 < δ 1. Stiff wellposedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2solution as δ → 0. The evolution of the limit is governed by a reduced hyperbolic system, the socalled equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff wellposedness—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 twophase flow in a porous medium and show stiff wellposedness 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 = {0022247X}, keyword = {hyperbolic system,relaxation,wellposedness,asymptotic expansion,equilibrium limit,speed condition}, language = {eng}, number = {2}, pages = {497532}, publisher = {Elsevier}, series = {Journal of Mathematical Analysis and Applications}, title = {Hyperbolic systems with relaxation: characterization of stiff wellposedness and asymptotic expansions}, url = {http://dx.doi.org/10.1006/jmaa.1999.6400}, volume = {235}, year = {1999}, }