Arnoldi shift-invert method
 

Dear Community,

I am doing some numerical linear stability analysis for fluid flows and taking advantage of
the shift-invert Arnoldi method but I am confused about something.

I know my "correct" eigenvalues have to be "around" my shift parameter but I
am not sure what "around" actually means, ie how close my eigenvalues need to
be the the shift parameter.

For example, I am look for real eigenvalues to determine when instability
occurs. If I have a shift parameter equal to one, should I believe I have an
instability if my first real eigenvalue is of the order
10^(-2)? Or should I only believe those that are of order unity, and assume
the others are spurious?

Thank you for your advice,
Archer

hi archer
 

As I know, the closer the invert parameter is to the real eigenvalue, the faster the eigen-equation can be solved. The converged eigenvalue has nothing to do with the invert parameter.

Thank you very much for your reply. A mixture of additional reading and testing brought me to that conclusion too. However, there is still something slightly puzzling me:

For same cases, I am getting different eigenvalues for different shift parameters. For example, at a given Reynolds number my leading eigenvalue may be small and negative for one value of the shift parameter and small and positive for another. This means one shift parameter says my flow is stable and another says it's unstable.

Do you think I should I therefore assume that this flows example is stable (but close to the critical Reynolds number) and that small numerical fluctuations/artefacts are to blame for this apparent contradiction? And that, for a truly unstable flow the leading (real part) eigenvalue should be positive regardless of the shift parameter? Or should I believe the result predicted when the shift parameter is close to the predicted real part of the leading eigenvalue?

Your attention to my message is very much appreciated,
Archer

Hi, in my opinion the invert parameter is not supposed to affect the result so much. Have you considered refining the mesh and try again?

Yes, that has crossed my mind and with your advice I think may well be the way forward.

Thank you very much.