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I plan to write one post in the series Fluid Flows and Infinite-Dimensional Manifolds about the nonlinear aspects of the Burgers equation. This equation turns up as the geodesic equation of the diffeormorphism group of the circle.
It is also the simplest example of a nonlinear equation, and it has solutions in closed forms. So, when Isaac Held writes about the fruit fly of climate models, the Burgers Equation is kind of the bacterium of nonlinear partial differential equations. If you are a coder without much experience in numerical mathematics and programming, this is a great place to start.
I would like to compare the solution in closed form in one (spatial) dimension to the approximation using the Fourier-Galerkin approximation. If we could get an online animation of the solutions, that would be a big big plus. I will try to get this done during the next two months, but cannot promise anything.
This thread is for anyone who would like to try to program this him/herself. Here are my thoughts about how this should be done so far:
Java is a pain when you need complex numbers already, because it does not come with this datatype and it does not have operator overloading and structures. So I won't use it.
C# shares one problem with Java, namely that there are no libraries for number crunching available, those are all in Fortran or in C++, so I won't use it either (or any language from the .NET platform),
Scala's overall tool chain was really disappointing when last I looked, so I won't use that either,
closed source software like Matlab or Mathematica should not be used in science,
Python is a toy, it is an interpreted scripting language which lacks a lot of concepts (type safety, for example) that would make it eligible for large software projects like Java or C++,
FORTRAN is known in academia only, at least I think so. Most coders out there won't know it, which includes me. And I am not inclined to spend months to learn it.
So I will try to use C++, but am still undecided which libraries I should use. Plus I have to invest some work to get fluent in C++, which I haven't used for a decade.
If you think I'm wrong and it is much easier to get the job done in a different way, we can talk about that here. If you agree with me and would like to help me, we can also talk about that here. Right now I am looking for the best way to include libaries for FFT and for the modified Bessel functions into an existing project, but I haven't checked in anything yet to our repository.
If there are several successful solutions to the posed problem, I will probably write a blog post about all of them, comparing them.
I will start today with stubs for said blog post and a page for C++ (doesn't exist right now), and will notify you here every other week or so, so that you can check my progress :-)