Incomplete Embedded Method for Order 3

[warisa-r]

This pull request introduces an incomplete implementation of an embedded Runge-Kutta method for order 3. The focus here is on utilizing the optimal b values for the method. The branch is based on PERK3, which is currently open on the main branch of Trixi.

Key changes include:

• Implementation of TrixiConvexClarabelExt for solving the convex optimization problem using Clarabel.
• Creation of methods_embedded_PERK3.jl, which contains the new struct of this method and the integration algorithm.
• Modification of TrixiNLsolve.jl to allow an extra argument for num_stage_evals to be passed when the number of stages (s) is not equal to the number of evaluations (e).
• Added two example files, based on what we discussed at the last meeting. I'm not entirely sure if this matches what you intended, so please review.

Let me know if there are any issues with the logic of the integrator or any other aspects of this implementation!

[github-actions]

Review checklist

This checklist is meant to assist creators of PRs (to let them know what reviewers will typically look for) and reviewers (to guide them in a structured review process). Items do not need to be checked explicitly for a PR to be eligible for merging.

Purpose and scope

• The PR has a single goal that is clear from the PR title and/or description.
• All code changes represent a single set of modifications that logically belong together.
• No more than 500 lines of code are changed or there is no obvious way to split the PR into multiple PRs.

Code quality

• The code can be understood easily.
• Newly introduced names for variables etc. are self-descriptive and consistent with existing naming conventions.
• There are no redundancies that can be removed by simple modularization/refactoring.
• There are no leftover debug statements or commented code sections.
• The code adheres to our conventions and style guide, and to the Julia guidelines.

Documentation

• New functions and types are documented with a docstring or top-level comment.
• Relevant publications are referenced in docstrings (see example for formatting).
• Inline comments are used to document longer or unusual code sections.
• Comments describe intent ("why?") and not just functionality ("what?").
• If the PR introduces a significant change or new feature, it is documented in NEWS.md with its PR number.

Testing

• The PR passes all tests.
• New or modified lines of code are covered by tests.
• New or modified tests run in less then 10 seconds.

Performance

• There are no type instabilities or memory allocations in performance-critical parts.
• If the PR intent is to improve performance, before/after time measurements are posted in the PR.

Verification

• The correctness of the code was verified using appropriate tests.
• If new equations/methods are added, a convergence test has been run and the results
  are posted in the PR.

Created with ❤️ by the Trixi.jl community.

[warisa-r]

Close so this stops sending notifications. It's from reaching the point of things being ready for review anyway.