Tau method for high-order equation,

[Guangpu Zhu]

Dear Dedalus experts,

 

         I am trying to solve a fourth-order equation in a rectangular domain with periodic conditions in the x-direction and Neumann conditions in the z-direction. The equation is like:

Here, w can be viewed as an intermediate variable.  We consider Neumann boundary conditions for both \phi and w in the z-direction. 

 

To impose 4 boundary conditions, I introduce four tau variables using the tau method introduced here, and the resultant equations are:

I further simplify these four equations into two and then implement them in Dedalus. However, I found that the obtained results are not desired. After carefully checking the parameters and implementation, I think maybe my understanding of the Tau method is not correct but I don't where I am wrong. 

I would greatly appreciate it if any experts can help me.

Best,

Guangpu

[Daniel Lecoanet]

Hi Guangpu,

I think you have pretty much the right idea… However, tau_w1 and tau_phi1 need to be multiplied by a z-dependent term, and all four tau variables should be x-dependent.

Daniel

On Apr 24, 2024, at 9:55 AM, Guangpu Zhu <zhug...@gmail.com> wrote:

Dear Dedalus experts,

 

         I am trying to solve a fourth-order equation in a rectangular domain with periodic conditions in the x-direction and Neumann conditions in the z-direction. The equation is like:

<Equation.png>

Here, w can be viewed as an intermediate variable.  We consider Neumann boundary conditions for both \phi and w in the z-direction. 

 

To impose 4 boundary conditions, I introduce four tau variables using the tau method introduced here, and the resultant equations are:

<Equation_Tau.png>

I further simplify these four equations into two and then implement them in Dedalus. However, I found that the obtained results are not desired. After carefully checking the parameters and implementation, I think maybe my understanding of the Tau method is not correct but I don't where I am wrong. 

I would greatly appreciate it if any experts can help me.

Best,

Guangpu

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<Equation.png><Equation_Tau.png>

[Guangpu Zhu]

Dear Prof.  Lecoanet,

         I am sorry for the typos in the above equation, the correct version I implemented is

Then I substitute the first and third equations into the second and fourth ones, respectively, which yields

We further introduce the intermediate vectors G_w and G_{\phi}, like 

We eventually get the following equation

In Dedalus, they are implemented as

-------

tau_ph1 = dist.Field(name='tau_ph1', bases=xbasis)
tau_ph2 = dist.Field(name='tau_ph2', bases=xbasis)
tau_w1 = dist.Field(name='tau_w1', bases=xbasis)
tau_w2 = dist.Field(name='tau_w2', bases=xbasis)

ex, ez = coords.unit_vector_fields(dist)
lift_basis = zbasis.derivative_basis(1)
lift = lambda A: d3.Lift(A, lift_basis, -1)
grad_ph = d3.grad(ph) - ez*lift(tau_ph1)
grad_w = d3.grad(w) - ez*lift(tau_w1)

problem = d3.IVP([ph, w, tau_ph1, tau_ph2, tau_w1, tau_w2], namespace=locals())
problem.add_equation("dt(ph) - div(grad_w) + lift(tau_w2) = 0")
problem.add_equation("w + div(grad_ph) - lift(tau_ph2) + ph = ph**3 ")
problem.add_equation("dz(ph)(z=-Lz/2) = 0")
problem.add_equation("dz(ph)(z=Lz/2) = 0")
problem.add_equation("dz(w)(z=-Lz/2) = 0")
problem.add_equation("dz(w)(z=Lz/2) = 0")

----------

Note that in the above code snippet, we consider the Neumann boundary conditions for both \phi and w:

Is my understanding of the tau method for this fourth-order equation right?  Thanks in advance.

Best,

Guangpu

[Keaton Burns]

Hi Guangpu,

Nothing is jumping out to me as incorrect about your problem formulation — can you describe a little more what issues you’re seeing? Is the system singular / blowing up, or just giving what seems to be a wrong result?  One problem could be that the w equation is purely algebraic, so the Dedalus timestepper is essentially doing a single fixed-point iteration on it, rather than something like a Newton solve, each step. You may need to check that you’re starting with consistent initial conditions (they satisfy the algebraic equation) and use small timesteps to keep this stable/accurate.

Best,

-Keaton

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[Guangpu Zhu]

Hi,  Keaton,

           Thanks for your kind reply. In my previous verification, I coupled this equation with the Navier-Stokes equation and failed to get the reference solution. Now I only implement this equation in Dedalus and compared the solution with that obtained by COMSOL. These two results agree very well.  Hence, I believe this implementation is right and my previous implementation (with hydrodynamics) is wrong. Thanks.

Best, 

Guangpu