MPDeC methods

[SKopecz]

Implementation of the MPDeC methods

• MPDeC for Conservative PDS
• MPDeC for non-conservative PDS
• Adaptive MPDeC
• Support sparse arrays
• Support tridiagonal matrices
• Nonautonomous MPDeC (out-of-place)
• Nonautonomous MPDeC (in-place)
• Testing
• Performance

This closes #42.

[codecov-commenter]

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Codecov Report

Attention: Patch coverage is 97.01087% with 11 lines in your changes missing coverage. Please review.

Project coverage is 97.76%. Comparing base (931ac70) to head (ea54eef).

Files with missing lines	Patch %	Lines
src/mpdec.jl	97.01%	11 Missing ⚠️

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Additional details and impacted files

@@            Coverage Diff             @@
##             main     #146      +/-   ##
==========================================
- Coverage   97.92%   97.76%   -0.17%     
==========================================
  Files           7        8       +1     
  Lines        1641     2009     +368     
==========================================
+ Hits         1607     1964     +357     
- Misses         34       45      +11     

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[SKopecz]

This should be ready for review 🙂.

Some additional comments:

• Currently MPDeC schemes up to order 10 are available.
• The quadrature nodes and weights are currently given as Float64s. Should we use algebraic expressions?
• Do we need tests that check the order of higher order MPDeC schemes? We probably would need BigFloats for this?
• Once this is merged, we can use MPDeC schemes in the benchmarks.
• Please find more specific comments below.

[JoshuaLampert]

The Downgrade job might be fixed by SciML/SimpleNonlinearSolve.jl#168 once this is released.

[JoshuaLampert]

JuliaRegistries/General#127812 has been merged. Could you please try if this fixes Downgrade by rerunning this job (I don't have the rights)?
Edit: Not necessary anymore. Looks like we need to bump the compat bounds to fix Downgrade as it is done in #153.

[JoshuaLampert]

This is really nice! Also cool to see that we can reproduce the order with Double64 also for the high-order schemes. I only have a few minor comments.

[ranocha]

Thanks a lot! I just have minor suggestions. This is ready to be merged when Joshua's comments are resolved as well.

[SKopecz]

In the original paper the MPDeC schemes depend on two parameters, $K$ and $M$. For the MPDeC scheme to be of order $K$ the parameter $M$ must be chosen sufficently large. In the code we use the smallest possible $M$ to obtain order $K$. The parameter M in the code represents exactly $M$. At the points in question, we now use M_plus_1 to have a consistent meaning of M throught the code.

[SKopecz]

The only failing test is CI / finish (pull_request).

Any idea how to fix this?

[JoshuaLampert]

The only failing test is CI / finish (pull_request).

Any idea how to fix this?

We also see this issue in other repos. Coveralls seems to be broken at the moment. I think we can just ignore it here.

[SKopecz]

Then we're ready to merge, right? Do we keep the version number "0.2.10-DEV"?

[ranocha]

Thanks