How to mitigate ringing?

[nathan magnan]

Hello everyone,

I'm working on a 1D problem with a background flow that has a sort of "smooth square function" shape.

I've tried to represent it in Dedalus. 

Unfortunately, I run into 3 problems, which I think are all connected:

1) The mode amplitude declines very slowly with mode number (it's still an exponential, but a very slow one: I would need to include 1024 modes to get good precision).

2) I see some ringing for the derivative in the plateau of the square function

3) I see some ringing for the derivative at the boundaries of the domain

I suppose it makes sense that a spectral code would struggle to represent something with sharp features. However, my issues persist even when I smooth the sides of the square to the point where they each occupy 25% of the domain.

I was wondering if someone has encoutered a similar issue before, and found a workaround?

Thanks for your help !

Nathan

PS: I attach a minimal working example, so you can play around.

[Jeffrey S. Oishi]

Hi Nathan,

I've only looked quite quickly, but it looks like you're only smoothing the slopes of the transition regions, leaving the point-wise jumps. This will never converge well as it still has singular derivatives. You should consider building your entire shape out of C-infinity functions. I'd suggest erf(); we use it to build smoothed versions of sharp structures with good results all the time.

Jeff

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[nathan magnan]

Hi Jeff,

Thanks for the quick answer.

I'll have a look at erf, but I did try to to match every derivative to create a C-infinity function.

Specifically, I used integral(ds, a -> x, 1 / (s - a)(b - s)) / integral(ds, a -> b, 1 / (s - a)(b - s)) as my smoothing function.

Best regards,

Nathan

[nathan magnan]

Hi Jeff,

I've run a test with erf, and I see the same thing:

There is one difference however: the ringing disappears above a certain smoothness, and the transition seems pretty sharp:

Between those two pictures, I only doubled the width of the transition region.

Equivalently, I can make the ringing disappear by increasing resolution by a factor two, but I'd prefer not to pay that cost.

And to be clear, there is still a bit of ringing, but the amplitude of the main oscillation is 10^(-8), which I can work with.

Best regards,

Nathan

[Jeffrey S. Oishi]

Hi Nathan,

Another issue is boundary conditions. You also need enough space in your domain for erf to be zero to machine precision, assuming you enforce zero boundary conditions.

 

There is one difference however: the ringing disappears above a certain smoothness, and the transition seems pretty sharp:

 Yes, that's expected for spectral accuracy.

Equivalently, I can make the ringing disappear by increasing resolution by a factor two, but I'd prefer not to pay that cost.

Alas, this is the fundamental tragedy of computational science.

Jeff