The stability of a linear hyperbolic problem

[dRic2]

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

 

[gtoguy97]

my question.What you are missing is that the condition for stability depends not only on the norm but also on the constant $c_t$. Different norms can result in different constants, and this in turn affects the stability of the method. For example, if the norm used to evaluate the condition is the Euclidean norm, then it is possible to show that the constant $c_t$ must be less than or equal to one for the method to be stable. If a different norm is used, then the constant will be different. This means that the same method can be stable with respect to one norm, but not another.