I am not too familiar with the ApproxFun “style” of numerical analysis. From what I gather relies on doing Chebyshev or Fourier approximations which sound really nice since they can have spectral convergence. However, there must be some cons to keep in mind if they aren’t the “standard go-to methods…

Have a look at the ChebFun docs which are extensive, ApproxFun is very similar. http://www.chebfun.org/

[image] mauro3: Have a look at the ChebFun docs which are extensive, ApproxFun is very similar. Is there a comparison or benchmarks to other methods somewhere in there? I’m not finding one.

Really hard to answer this question in a vacuum; pros/cons depend strongly on what precisely you are doing. Solving a PDE? Finding roots? Interpolating?

[image] stevengj: Really hard to answer this question in a vacuum; pros/cons depend strongly on what precisely you are doing. That’s why I’m probably looking for a good read: a paper/blogpost/book chapter which goes into some detail. I’m usually numerically solving ODEs/SDEs/(S)PDEs. But tha…

“Chebyshev and Fourier Spectral Methods” by Boyd is a pretty good textbook on the sort of thing that ApproxFun and Chebfun automate, and can be downloaded free of charge online. Trefethen has a couple of books on this sort of thing, too. My short answer is that spectral methods of this sort are ex…

Thanks! That’s just the kind of overview with references I was looking for. Taking a quick look at Chapter 9 of the Boyd book, their relation to ODEs/PDEs is they give a different way to discretize space which converges exponentially (in some problems), and then you can throw this discretization in…

It’s more a question of basis functions than discretization, but yes, you can then use standard ODE integration schemes like Runge-Kutta. And the key question is not so much the PDE as the shape of the domain and the smoothness/regularity of the PDE coefficients, boundary conditions, and solutions.

Boyd is more collocation focused, whereas ApproxFun poses equations in coefficient space to get structured (e.g. Banded) operators and linear complexity. Combining ApproxFuns approach with traditional time stepping code is possible, but hasn’t been done yet. There’s a placeholder package ApproxFun…

[image] dlfivefifty: Combining ApproxFuns approach with traditional time stepping code is possible, but hasn’t been done yet. There’s a placeholder package ApproxFun/SpectralTimeStepping.jl that is meant to be the home for that. It currently has some simple time steppers built on top of Approx…

I started the following issue: Integrate with time stepping codes opened 04:07AM - 18 Nov 16 UTC (UTC) [dlfivefifty] dlfivefifty This should be a home…

Pros/Cons of ApproxFun

Specific Domains Numerics

ChrisRackauckas November 18, 2016, 3:50am 10

Looking at your SpectralTimeStepping.jl codes, the operators that ApproxFun makes end up working a lot like functions, right? I’m wondering what exactly would have to be done to support solving PDEs defined by ApproxFun in the DiffEq common solve interface, do you know? Maybe we should take this to an issue.

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Pros/Cons of ApproxFun

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