I The stability of a linear hyperbolic problem

  • Thread starter dRic2
  • Start date

dRic2

Gold Member
540
94
Hi, I'm reading a book about numerical models for PDE and it says that a method is said to be stable if this condition holds:
$$|| \mathbf u^{n+1} || \le c_t || \mathbf u^{n} ||$$
where ##c_t## is a constant greater than zero, and ##u## is the numerical solution to the problem. (In particular I'm studying transport equations). I think that this condition is equivalent to the following:
$$||\mathbf {\Phi}|| < c_t$$
where ##\Phi## is the "iteration matrix" (##\mathbf u^{n+1} = \mathbf {\Phi} \mathbf u^{n}##).

There is something that bothers me very much in this definition... Here it seems to me that any norm will do the job. I mean, I know that in ##R^N## all the norms are equivalent so if the condition happens to be true for a particular norm then it will be true for all the other norms... But in the book the author says that stability is linked to the norm I chose to evaluate it; in particular a problem could be stable with respect to a certain norm A, but not B.

What am I missing ?

PS: I hope I explained myself properly, otherwise let me know if you have problems to understand
 

Want to reply to this thread?

"The stability of a linear hyperbolic problem" You must log in or register to reply here.

Related Threads for: The stability of a linear hyperbolic problem

Replies
0
Views
2K
Replies
6
Views
899
Replies
1
Views
738
  • Posted
Replies
0
Views
3K
Replies
3
Views
3K
Replies
1
Views
784
Replies
0
Views
1K
Replies
0
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top