Help for defining the tau method and grid space initial condition.

[Ritik Dubey]

hey all, 

I am a 2nd yr student of IISER Kolkata, India and I am trying to learn this as an intern in IIT Kharagpur, India.

I am not able to understand the followings,

1. u['g']=? , how it is defined , why taking any arbitrary function, what is the concept.

In periodic shear 2D (IVP) example

# Initial conditions # Background shear u['g'][0] = 1/2 + 1/2 * (np.tanh((z-0.5)/0.1) - np.tanh((z+0.5)/0.1)) # Match tracer to shear s['g'] = u['g'][0] # Add small vertical velocity perturbations localized to the shear layers u['g'][1] += 0.1 * np.sin(2*np.pi*x/Lx) * np.exp(-(z-0.5)**2/0.01) u['g'][1] += 0.1 * np.sin(2*np.pi*x/Lx) * np.exp(-(z+0.5)**2/0.01)

 

2. tau method, 

first i thought it is to impose the condition then, i saw some examples where it is using even when no condition is needed to impose on the equation. also don't get the below red marked lines.

# Tau polynomials tau_basis = xbasis.derivative_basis(2) p1 = dist.Field(bases=tau_basis) p2 = dist.Field(bases=tau_basis) p1['c'][-1] = 1 p2['c'][-2] = 2

If someone can provide me some material or video lec. it will be really beneficial, i have already gone through the tutorial and example section of the dedalus.

Best,

- Ritik

[Keaton Burns]

Hi Ritik,

1. I’m not quite sure what your specific question is — this code is setting the initial conditions of the velocity vector in grid space. This is set using numpy array expressions using the grid-coordinate array z.
   

2. I’d suggest taking a look at the tau page in the documentation. The tau terms are used to enforce boundary conditions. The code you’ve shown is one way to do this with Chebyshev methods, by explicitly constructing the tau polynomials, but the suggested route (which should be in all the examples) is to use the “Lift” operator.

-Keaton

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[Ritik Dubey]

Thank you for your response sir, but here one thing I wanna know, Is tau method a concept for dedalus software or it's a mathematical concepts of behind the scene.
If it is mathematical, then where they taught in math. I mean Topic of Math for studying it.

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[Keaton Burns]

Hi Ritik,

It’s a mathematical approach to interpreting the modifications needed to apply boundary conditions in terms of changes to the underlying PDE. You can find more, including lots of the original math papers on the topic, by googling for "tau spectral method” or similar.

Best,

-Keaton

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[Ritik Dubey]

Thanks you all

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