Eigenvalue problem and initial-value problem?

[jollage]

Hi all,

I want to ask a question about the eigenvalue problem (EVP) and the initial value problem (IVP).

Let's say we are solving this linear equation \frac{\partial u}{\partial t}=\mathcal{L}u, the operator L is dependent on some parameters like Reynolds number.
I first check the eigenvalue of the equation at a specific parameter set.

My question is if the eigenvalue at this specific parameter is positive (meaning the system is unstable), now I solved the IVP counterpart \frac{u^{n+1}-u^n}{dt}=\mathcal{L}u with an ARBITRARY initial condition, will this initial condition amplifies over all the time history (especially at the early phase, because I know asymptotically the system is governed by the positive eigenvalue)?

Thanks in advance

Jo

 

[the_wolfman]

Hi all,

I want to ask a question about the eigenvalue problem (EVP) and the initial value problem (IVP).

Let's say we are solving this linear equation \frac{\partial u}{\partial t}=\mathcal{L}u, the operator L is dependent on some parameters like Reynolds number.
I first check the eigenvalue of the equation at a specific parameter set.

My question is if the eigenvalue at this specific parameter is positive (meaning the system is unstable), now I solved the IVP counterpart \frac{u^{n+1}-u^n}{dt}=\mathcal{L}u with an ARBITRARY initial condition, will this initial condition amplifies over all the time history (especially at the early phase, because I know asymptotically the system is governed by the positive eigenvalue)?

Thanks in advance

Jo

Assuming that you're initial perturbation is sufficiently random, it will kick off all the unstable modes of the system. However the most unstable mode will grow fastest and overtime it will dominate the solution.
Thus the procedure you describe will asymptotically yield the eigenvector associated with the largest eigenvalue.
Does that make sense?

 

[jollage]

Assuming that you're initial perturbation is sufficiently random, it will kick off all the unstable modes of the system. However the most unstable mode will grow fastest and overtime it will dominate the solution.
Thus the procedure you describe will asymptotically yield the eigenvector associated with the largest eigenvalue.
Does that make sense?

Hi the_wolfman,

I have figured out what's going on here.

I have originally many negative eigenvalues and one positive eigenvalues, so initially the energy of the state should decay before asymptotically increase. Previously, I failed to look at a sufficiently long time (I observe the energy "always" decays), so I thought I did something in the numerics. Now I fixed it, what I have to do is to run a very long time before I would see the decay of the energy.

Thank you anyway.

Jo