Convolution/integration operation

[alexis....@gmail.com]

Dear all,

I would like to implement the following term in Dedalus, for a 1+1PDE, with $u(x,t)$:

$$f(x,t) = \int_{-\infty}^x u(y,t) dy, $$

which can be seen as a convolution of u with the Heavyside step function.

I could not find such an operation, as 'integ' does an integration over the entire space, and I don't know how to implement a Heavyside function which needs to depend on two variables, $x$ and $y$.

Thanks in advance for your help!
Alexis Arnaudon

[Keaton Burns]

Hi Alexis,

I think the simplest way to do this is to expand the problem variables to include f and enforce the relation through a differential equation like “dx(f) - u = 0”, along with a boundary condition to set the constant of integration.

Best,

-Keaton

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[arnaudon alexis]

Hi Keaton,

this is a really good idea! I tried but I failed so far... When I use what you sugested I obtain the following error after the first timestep:

/python3.5/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:145: MatrixRankWarning: Matrix is exactly singular
  warn("Matrix is exactly singular", MatrixRankWarning)

Also, I am using periodic boundary conditions, which may be the issue here. I don't see how to solve this problem, which seems to be coming from the fact that I define f via its derivative (if I remove the derivative on f, the code does not crash).

Thanks a lot for your help!

[Daniel Lecoanet]

Hi there,

I think you may not be setting the gauge on f.  For instance, taking f->f+4 does not change your problem.  To set the gauge, you can do something like

problem.add_equation("dx(f) - u = 0", condition="nx != 0")

problem.add_equation("f = 0", condition = "nx == 0")

Daniel

On Tue, Mar 7, 2017 at 9:28 AM, arnaudon alexis <alexis....@gmail.com> wrote:

Hi Keaton,

this is a really good idea! I tried but I failed so far... When I use what you sugested I obtain the following error after the first timestep:

/python3.5/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:145: MatrixRankWarning: Matrix is exactly singular
  warn("Matrix is exactly singular", MatrixRankWarning)

Also, I am using periodic boundary conditions, which may be the issue here. I don't see how to solve this problem, which seems to be coming from the fact that I define f via its derivative (if I remove the derivative on f, the code does not crash).

Thanks a lot for your help!

On Monday, March 6, 2017 at 3:11:41 PM UTC, Keaton Burns wrote:

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[Ben Brown]

Followup:

On a periodic domain you can't sent boundary conditions.  I'm not sure if "add_bc()" throws an error on a periodic Fourier domain, or if it causes a singular matrix.  And you may not be doing this anyways, already.  But something to be aware of.

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[arnaudon alexis]

Thanks a lot, Daniel, it works well! I'll make sure I'll cite your paper and acknowledge your help! Dedalus is a really nice tool, keep developing it!

Ben, I tried what you said, and using add_bc with Fourier modes does not work, Dedalus complains about it.

All the best!

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