Left/Right eigenmode orthogonality and tau terms

[Zach Williams]

Hello Dedalus users,

I'm working on some research that involves a deep dive into eigenmode solutions of various PDE models. I've had success with this in the past but am running into issues that I have slight reasons to believe are connected to tau terms.

To be specific: I'm solving EVPs for both left and right eigenmodes to do subsequent analysis, but as a first sanity check it's good to ensure that the left and right modes are orthogonal to one another. I am able to achieve this when testing on equations that do not contain tau terms, but it seems to fail in systems that do. Please see the two attached scripts.

- temp_test.py is a simple modification of the Dedalus example for the Mathieu equation (no tau terms). If you run that, it will print out the identity matrix, which confirms left/right orthogonality

- temp_test2.py is the same thing except for the wave on a string example (which contains tau terms). In that case, the matrix printed out is most certainly not the identity.

I don't have a great handle on tau terms so it's possible I'm missing something or it could be something else entirely. My hope is to implement this is considerably more complicated systems that will use many tau terms, so I appreciate any input you all have!

Thanks so much for your time.

Cheers,

Zach Williams

[Calum Skene]

Hi Zach,

Is the issue the way the matrix is calculated? The eigenvectors are the columns of solver.eigenvectors, and so I think it is

print(np.transpose(solver.modified_left_eigenvectors).conj()@solver.right_eigenvectors)

that should give the identity (no transpose on right_eigenvectors). That works for me for both tests.

The Mathieu script works even if you transpose right_eigenvecors since right_eigenvectors is an orthogonal matrix for that example.

Hope this helps,

Calum