Boundary condition defined in Fourier space

[Sadokat]

Dear All,

I am working with Dedalus v3, my vector field u is defined using a mixed basis RealFourier in x and y, and Chebyshev in z. Mathematically I derived the Robin type boundary condition in Forier space, something like: 

∂z û (z) + k_⊥ û (z) = 0, on z=Lz.

where k_⊥ = sqrt(kx² + ky²). 

Are there any approaches in Dedalus for operators related to fractional Laplacians (−Δ_⊥)^{−1/2}? 

 I would appreciate some guidance on how to properly formulate this kind of  boundary condition. 

with best regards,

Sadokat

[Sadokat]

Dear All,

I believe I’ve resolved my issue, but I’d really appreciate confirmation that this is a correct approach in Dedalus v3.

My goal is to impose boundary condition by constructing a vector field directly in coefficient space.   I do:

u_z = dist.VectorField(coords, basis=(xbasis,ybasis))

u_z['c'] = -k_perp * u['c'](z=Lz) ### this is just a short explanation of the idea

and then use the physical domain to express the boundary condition:

problem.add_equation("dz(u)(z=Lz) = u_z") # Is this correct?

I followed the implementation of the real Fourier transform in transform.py and defined:

kx = xbasis.wavenumbers[:, None, None]

ky = ybasis.wavenumbers[None, :, None] 

kperp = np.sqrt((2*np.pi*kx/Lx)**2 + (2*np.pi*ky/Ly)**2)

Is this the correct way to construct    k_⊥?

I would really appreciate it if someone could confirm whether this is the recommended approach.

And thank you to the Dedalus developers for such an amazing framework!

Best regards,

Sadokat

[Daniel Lecoanet]

Hi Sadokat,

I’m not sure if that is doing what you desire. The issue is that u_z is not updated. I think what you’re trying would work if you include the u_z[‘c’] assignment in the timestepping loop (make sure you use a multistep timestepper). There will also be a time lag in the BC as u_z is evaluated on the previous timestep as dz(u). Unfortunately I do not believe we currently have a way of evaluating k_perp implicitly (on the LHS of the equation).

Daniel

On Feb 6, 2026, at 08:03, Sadokat <smaliko...@gmail.com> wrote:



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[Sadokat]

Dear Daniel,

Thank you very much for your response.

I am updating u_z = funct(u) inside the time-stepping loop, and I verified that its value is being updated correctly. Based on this, I moved the term involving k_perp to the RHS:

problem.add_equation("dz(u)(z=Lz) = u_z")

With this approach, it appears to be working as intended.

Thank you once again for your help — I really appreciate it.

Best regards,
Sadokat