Add TS example ex51

[ranocha]

I wanted to use some time-stepping methods from PETSc.jl since we were asked to use a high-order Gauss method for a paper revision. It turned out that PETSc implements them but only supports sparse Jacobians, while we need a dense Jacobian. Thus, I cannot use PETSc.jl 😢

Nevertheless, I had some code setting up an additional example. Thus, I thought it would be nice to share this in PETSc.jl.

Disclaimer: I used Codex to help me to write some code. I reviewed the code and updated it in several places.

[boriskaus]

Thanks a lot, this is a great addition!

Is it not possible to represent dense arrays in a sparse format? Or is that slow?

[ranocha]

I would have expected to get massive performance problems when trying to represent a dense $$2^8 \times 2^8$$ matrix as a sparse matrix. But I didn't try it out, I have to admit...

[ranocha]

Could you maybe enable only squash-merge to avoid cluttering the history of the main branch?

[boriskaus]

Could you maybe enable only squash-merge to avoid cluttering the history of the main branch?

If you tell me how to do that, sure

[ranocha]

Go to the settings of this repo (likely https://github.com/JuliaParallel/PETSc.jl/settings) and disable the options "Allow merge commits" and "Allow rebase merging" (or give me admin rights so that I can do that).

[boriskaus]

thanks - done.