Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)
(1997) In Advances in Differential Equations 2(4). p.643-666- Abstract
- The paper deals with the Cauchy problem for the linear constant-coefficient strongly hyperbolic system $u_t+Au_x=({1}/{delta})B$.
The critical case where $B$ has a nontrivial nullspace is investigated. Under suitable assumptions on the matrices $A$ and $B$ the convergence in $L_2$ as $delta o 0$ of the solution $u(·,t,delta)$ of the Cauchy problem is proved. Then the evolution of the limit function ${check u}(·,t)$ as a solution of the Cauchy problem for the strongly hyperbolic system without zero-order term is described. This system is called an equilibrium system and a method for obtaining it is given. A concept of stiff well-posedness is introduced that ensures solution estimates independent of $0<deltale 1$. It... (More) - The paper deals with the Cauchy problem for the linear constant-coefficient strongly hyperbolic system $u_t+Au_x=({1}/{delta})B$.
The critical case where $B$ has a nontrivial nullspace is investigated. Under suitable assumptions on the matrices $A$ and $B$ the convergence in $L_2$ as $delta o 0$ of the solution $u(·,t,delta)$ of the Cauchy problem is proved. Then the evolution of the limit function ${check u}(·,t)$ as a solution of the Cauchy problem for the strongly hyperbolic system without zero-order term is described. This system is called an equilibrium system and a method for obtaining it is given. A concept of stiff well-posedness is introduced that ensures solution estimates independent of $0<deltale 1$. It is shown that for $2 imes 2$ systems the requirement of stiff well-posedness is equivalent to the well-known subcharacteristic condition. The theory is illustrated by examples. One of them shows that the assumption of strong hyperbolicity is essential. Lastly, some numerical experiments for a $3 imes 3$ system are carried out. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1224357
- author
- Schroll, Achim ^{LU} and Lorenz, Jens
- organization
- publishing date
- 1997
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Advances in Differential Equations
- volume
- 2
- issue
- 4
- pages
- 643 - 666
- publisher
- Khayyam Publishing, Inc.
- ISSN
- 1079-9389
- language
- English
- LU publication?
- yes
- id
- 024f397d-48d3-4b49-a2bf-15b3199710be (old id 1224357)
- date added to LUP
- 2008-09-02 11:41:56
- date last changed
- 2018-05-29 12:10:30
@article{024f397d-48d3-4b49-a2bf-15b3199710be, abstract = {The paper deals with the Cauchy problem for the linear constant-coefficient strongly hyperbolic system $u_t+Au_x=({1}/{delta})B$. <br/><br> <br/><br> The critical case where $B$ has a nontrivial nullspace is investigated. Under suitable assumptions on the matrices $A$ and $B$ the convergence in $L_2$ as $delta o 0$ of the solution $u(·,t,delta)$ of the Cauchy problem is proved. Then the evolution of the limit function ${check u}(·,t)$ as a solution of the Cauchy problem for the strongly hyperbolic system without zero-order term is described. This system is called an equilibrium system and a method for obtaining it is given. A concept of stiff well-posedness is introduced that ensures solution estimates independent of $0<deltale 1$. It is shown that for $2 imes 2$ systems the requirement of stiff well-posedness is equivalent to the well-known subcharacteristic condition. The theory is illustrated by examples. One of them shows that the assumption of strong hyperbolicity is essential. Lastly, some numerical experiments for a $3 imes 3$ system are carried out.}, author = {Schroll, Achim and Lorenz, Jens}, issn = {1079-9389}, language = {eng}, number = {4}, pages = {643--666}, publisher = {Khayyam Publishing, Inc.}, series = {Advances in Differential Equations}, title = {Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)}, volume = {2}, year = {1997}, }