Advanced

Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)

Schroll, Achim LU and Lorenz, Jens (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:
author
organization
publishing date
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
2016-04-16 06:24:09
@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 $0&lt;deltale 1$. 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},
}