Toy implementation for 1D Heat equation with time-dependent BCs in ultraspherical discretisation

[brownbaerchen]

This is a toy implementation of a pySDC-native implementation of a heat equation with time-dependent boundary conditions. I didn't minimize code-duplication because I thought that it might make it easier to understand what's happening when all relevant code is in the same file.

This is supposed to be a simpler version of #632, not sure if you would agree that it's simpler :D I realize it's maybe a little opaque. I thought I do this sort of minimal thing now and, if it seems reasonable, I can provide more explanations or code. Also, my thesis contains a lot of explanations for Chebychev methods that are very close to the implementation in pySDC.

The power of the Chebychev and ultra spherical methods regarding BCs is that you can put whatever you want in the lines of the boundary conditions. All you need to know is which lines of the systems of equations you need to put the BC in. You don't even need to transform, you put in the BCs in physical space. This is not much more difficult in ND than here and non-linear equations are no more difficult from a BC point of view. Of course, we would have to do IMEX splitting because we can't treat the non-linear term implicitly here.

I added a test that the boundary conditions are satisfied after time. It's not run in the CI, since this is a playground.
Let me know what you think about this.

[pancetta]

I appreciate the different angle, thanks a lot. For me not having a clue about ultra-mega-spectral methods this seems less "simple" then the "plain" FEM solution of #632 and this will limit us to IMEX methods for nonlinear problems (which is no big deal given that this is not the point of the whole demonstration).

[pancetta]

Unless this is too much work, would you be able to reproduce the test seen in #632 that demonstrate order reduction (and fixes by boundary lifting)?

[brownbaerchen]

I haven't looked into the lifting stuff, so no idea how difficult that is to implement. Anyways, here is a plot with the order reduction with 3 nodes. Is that similar to what you observe in FE?