Allow LinearSolve v3

[JoshuaLampert]

I just switched to KLUFactorization() instead of LUFactorization(). Let's see if CI passes. Do we generally prefer LUFactorization() over KLUFactorization()? So should we continue to search for a fix for LUFactorization() or should we change the default to KLUFactorization() at least for now to let the tests pass again (would be breaking though)?

Using KLUFactorization() instead of LUFactorization for sparse matrices fixes the pattern of the matrix changed error from LinearSolve.jl v3.
Closes #131, closes #135, closes #136, closes #137, closes #150, closes #151, closes #152.

[codecov-commenter]

⚠️ Please install the to ensure uploads and comments are reliably processed by Codecov.

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.92%. Comparing base (d8a3c61) to head (fa4b682).

❗ Your organization needs to install the Codecov GitHub app to enable full functionality.

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #153   +/-   ##
=======================================
  Coverage   97.92%   97.92%           
=======================================
  Files           7        7           
  Lines        1641     1641           
=======================================
  Hits         1607     1607           
  Misses         34       34           

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[JoshuaLampert]

This is ready for a review. Regular CI is passing. For the Downgrade test I'll see if I can debug this in another PR.

[JoshuaLampert]

To clarify, this PR is also only kind of a hotfix as in general we should be able to solve sparse problems with LUFactorization. If we want to merge this, we also need to think about refactoring https://skopecz.github.io/PositiveIntegrators.jl/stable/heat_equation_dirichlet/#Performance-comparison as we compare both (LUFactorization abd KLUFactorization) there.

[ranocha]

Thanks for all the work!

[ranocha]

To clarify, this PR is also only kind of a hotfix as in general we should be able to solve sparse problems with LUFactorization. If we want to merge this, we also need to think about refactoring https://skopecz.github.io/PositiveIntegrators.jl/stable/heat_equation_dirichlet/#Performance-comparison as we compare both (LUFactorization abd KLUFactorization) there.

We need to discuss this - the docs fail because of this (https://github.com/SKopecz/PositiveIntegrators.jl/actions/runs/14036192931/job/39294690579?pr=153#step:7:27)

[JoshuaLampert]

There are more and more errors, when we try to keep support of LinearSolve.jl v2, see https://github.com/SKopecz/PositiveIntegrators.jl/actions/runs/14127496106/job/39579796614?pr=153#step:8:425 for the option, where we require LinearSolve v2.39.1 as suggested by @ChrisRackauckas. I don't understand where the error comes from, but also don't want to spend the time to debug this. So let's just drop support for LinearSolve.jl v2.

[ranocha]

Thanks a lot!