# I The stability of a linear hyperbolic problem

#### dRic2

Gold Member
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

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