Implementing boundary conditions

[Saptarshi Dasgupta]

Hello,

I am new to Dedalus and I am solving a system of equations that is attached. I have already solved the system in two dimensions using a Fourier basis and now, I want to solve the system of equations in a semi-infinite box (periodic boundary conditions in the x direction and a fixed boundary in the y direction). I understand this implies using a Fourier basis in the x direction and a Chebyshev basis in the y direction.

I have a system of coupled partial differential equations in displacement (u), density fields (\rho and \phi respectively) and I need to implement boundary conditions (u = 0, grad (\rho)=0, grad (\phi) = 0 at the boundaries y=0 and y=Ny). 

I cannot understand how to implement the tau method for this case i.e. derive a first order form for my system of equations.  

I suppose I have to add tau fields corresponding to the number of boundary conditions I need to impose (in this case: two for u, two for phi, two for rho). 

The Rayleigh-Benard convection (2D IVP) problem example script defines lift as : 

lift= lambda A: d3.Lift(A, lift_basis, -1)

Whereas, the Poisson equation example script defines the lift operator slightly differently as : 

lift_basis = ybasis.derivative_basis(2)

lift = lambda A, n: d3.Lift(A, lift_basis, n)

I’m confused about how I should define the lift operator for my problem.

Finally, as stated earlier, I cannot understand how to re-cast my system of equations in a first order form so as to include the tau terms (like lift(tau_u1, -1) etc.

I understand that div(u), lap(u) also has to change while implementing in first order form, is a definition like this justified in my problem?

div(u) = trace(grad_u) and grad_u = d3.grad(u) + ey*lift(tau_u1)

Would be great if anyone could help me out. I’m attaching a working code of the system of equations in two dimensions (using Fourier basis) with this post.

Thanks in advance!

- Saptarshi Dasgupta,

Simons Centre for the Study of Living Machines,

National Centre for Biological Sciences, Tata Institute of Fundamental Research

[Keaton Burns]

Hi Saptarshi,

The Poisson example directly implements second-order taus as described in the docstring, but I’d suggest using the first-order form as is done in the Rayleigh-Benard example. Please take a look at the tau page here in the documentation, which covers the process if reducing your equations to first-order, and incorporating the first-order tau corrections in auxiliary variables as done in the RBC example.

Best,

-Keaton

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