Dear Leonardo,

I think such eigenvalues are strictly a result of the discretization and are not related to the operator/boundary conditions. Without knowing the equations exactly, in your case I guess that the eigenvalue parameter does not appear in the boundary conditions. When you discretize the problem, do you also include these boundary conditions explicitly? Then, for each boundary condition not involving the eigenvalue parameter, the resulting algebraic system contains an equation which is, again, free of the eigenvalue. Nevertheless, the eigenvalue solver will generate eigenvalues  for these equations too. These are exactly the spurious ones. 

To avoid this, I suggest incorporating implicitly in the discretization the boundary conditions that are not involving the eigenvalue parameter. This might work easier for some Galerkin type discretization. 

You may find the following interesting:

C.I. Gheorghiu, I.S. Pop, A modified Chebyshev-tau method for a hydrodynamic stability problem*, Approximation and Optimization, Proceedings of the International Conference on Approximation and Optimization, D.D. Stancu, Gh. Coman, W.W. Breckner, P. Blaga (eds.), Transilvania Press, Cluj-Napoca, vol. II (1997), pp. 119-126, with C.I. Gheorghiu. (https://docs.google.com/a/uhasselt.be/viewer?a=v&pid=sites&srcid=dWhhc3NlbHQuYmV8c29yaW4tcG9wfGd4OjExNjY2NmMyOGEzNDk0ZTY)

I.S. Pop, A stabilized Chebyshev-Galerkin approach for the biharmonic operator*, Bul. Stiint. Univ. Baia-Mare Ser. B, (now Carpathian J. Math.) Vol. XVI (2000), pp. 335 - 344. (https://docs.google.com/a/uhasselt.be/viewer?a=v&pid=sites&srcid=dWhhc3NlbHQuYmV8c29yaW4tcG9wfGd4OjMyZWI2YjJiNTNmY2I1ZmQ)

Good luck,

Sorin

Thank you @Pop

The imaginary eigenvalues at short wavelengths correspond to the Kelvin-Helmholtz instability. The inviscid differential problem becomes ill-posed if the shear layer is very thin. In such case you should add viscosity to filter the eigenvalues.

Dear Leonardo,

I have some experience in finding spectrum to Orr-Sommerfeld equation using the finite-difference approach and QZ algorithm. It appears that the spurios iegnevalues are not only the result of discretization of boundary conditions as stated by Iuliu. Please reffer to J. Gary, R. Helgason, 1970 A Matrix Method for Ordinary Differential Eigenvalue Problems, J. Comp. Phys. 5, 169-187. Eigenvalues can appear due to excessive approximation stencil as well (for example, in solution to Orr-Sommerfeld equation, the largest approximation stencil contains 9 mesh nodes). The authors noted that the spurious eigenvalues are very sensitive to the mesh resolution (and to approximation stencile size), so this can be one of the clues to eliminate them.

Kind regards,

Sergei

Dear Leonardo,

you can try to get rid of some spurious modes by using a complex mapping , see for instance:

1/ Appendix B ("The complex mapping function")

in

Kelvin waves and the singular modes of the Lamb-Oseen vortex

By Fabre, Sipp & Jacquin (2006)

J. Fluid Mech. 551, pp. 235-274

or

2/ Appendix

in

Kelvin-Helmholtz instability in the presence of variable viscosity for mudflow resuspension in estuaries

By Harang, Thual, Brancher & Bonometti (2014)

Env. Fluid Mech. 14, pp. 743-769

It's a bit empirical, but it is efficient - when it works :-)

I hope that helps.

Pierre

Dear all,

Just an adding that motivates to my previous answer: it is inspired by computing the spectrum for the Orr Sommerfeld equation (among others) in the standard case, as well as with boundary conditions emerging from a free surface of a thin film. I used a spectral method based on Chebyshev polynomails, as such methods are highly accurate and therefore well suited for approximating more eigenvalues than the most unstable mode. Of course, once the problem is discretized, one needs to use an algorithm for computing the algebraic eigenvalue problem (in this case, generalized: A V = \lambda B V, with A and B matrices and V an eigenvector corresponding to \lambda).

Coming back to the discretization: the standard Chebyshev tau approach gives spurious eigenvalues next to the ones that are approximated correctly (thumbrule: if N is the order of polynomials involved in the discretization, then the first about 0.6 * N eigenvalues are approximated accurately). When using either a Chebyshev Galerkin approach (for the standard Orr Sommerfeld equation), or a tau version that incorporates a priori the parameter free boundary conditions (for the second case) the spurious eigenvalues are not present anymore in the calculations. 

No meaning to impose an opinion, I just wanted to share my experience (well, some time ago...). It could be that I have misinterpreted the notion of  "spurious", but from my prospect I think these are EW's that simply have nothing to do with the original problem, thus not numerical approximations that are not accurate. 

Good luck further,

Sorin

May be you can find something useful in this paper and the references therein:

Hagan, J., & Priede, J. (2013). Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method. Journal of Computational Physics, 238, 210-216.

 https://www.researchgate.net/publication/235410545_Capacitance_matrix_technique_for_avoiding_spurious_eigenmodes_in_the_solution_of_hydrodynamic_stability_problems_by_Chebyshev_collocation_method

Article Capacitance matrix technique for avoiding spurious eigenmode...

Dear Leonardo,

It is likely that spurious eigenvalues appear because of your numerical approximation of boundary conditions not involving the eigenvalue as Sorin noted. Such spurious eigenvalues often can be infinite (they would have very large or, in contrast, close to zero numerical values when computed using standard generalized eigenvalue solvers). You can easily filter them out by making your code ignore any eigenvalues that are larger (smaller) than a set threshold. Unfortunately, spurious eigenvalues can also be finite, but they still can be filtered out without changing your chosen approximation method e.g. still using Chebyshev collocation method. It is characterized by a  fast (exponential) convergence for physical eigenvalues but the spurious eigenvalues keep changing as you vary the number of collocation points. This observation can be used to remove them. Try the following filtering procedure:

1. Compute the eigenvalue spectrum L_i, i=1,2,...N using N collocation points.

2. Compute the eigenvalue spectrum M_i, i=1,2,...,N+1 using N+1 collocation points.

3. Sort both spectra in the order of decreasing temporal amplification rate (real or imaginary part of eigenvalues, depending on how you define it).

4. Set some tolerance parameter TOL and start comparing the eigenvalues from the second set to those in the first set: e.g. choose M_1 and compare it with L_1, L_2, L_3 etc. Stop if abs(M_1-L_m)

Hello all--

If indeed the origin of the issue is that there is a boundary condition that does not include an eigenvalue, then this means that in the generalized problem that arises from discretization, Ax=lambda*Bx, (A and B are matrices and lambda is eigenvalue), there is a row of zeros in the matrix B corresponding to the boundary condition imbedded in A.  What I have done in these cases is take a multiple of the row in matrix A, and put that multiple into the matrix B.  This creates a known (but spurious) eigenvalue that equals the multiple, and you know precisely what it is.  You can verify this by changing the multiple, and then the eigenvalue moves (the physical ones do not).  Then, I rule it out.  I have only done this in problems where I have used simplistic finite differences to solve certain eigenvalue problems, so perhaps my suggestion is not relevant to your problem...but it seems that regardless of discretization scheme, the generalized problem will arise and so my approach could be relevant.

Additionally, I have also solved stability problems in which the interface speed itself corresponds to one of the eigenvalues.  These are also spurious eigenvalues, as such a speed would give an infinite interface height owing to the kinematic constraints applied at the interface.

Pardon me if my answer is off base relative to what has been said...my recent work has taken me a bit away from such numerical eigenvalue analyses and my info could be dated.

--Best,

--Steve

We also face this problem when we deal with a generalized eigenvalue problem when one of the matrices is singular. The Chebshev Collocation method is not stable and very sensitive, as we vary N we get different results..

The topic seems to be obsolete, but just a curios question to Tapan: spectrum to boundary layer problem is continuous (not even discrete) , so what do you mean by "the actual number of eigenvalues are very few "?

Dear Tapan, by "obsolete" I meant this particular tread, not the whole topic, sorry for confusing. My comment is related to your statement implying that the boundary layer spectrum is finite. But I guess you meant that the actual non-modal stability of a boundary-layer (or other external) flow is governed by the first several modes and you don't need to calculate full spectrum.

And thank you for the references!

You can check this paper. It may help you.

A pseudospectral approach applicable for time integration of linearized N-S operator that removes pole singularity and physically spurious eigenmodes.

Int J Numer Meth Fluids. 2019;91:473–486.

Leonardo Antonio De Araujo