PERK time integration schemes' documentation

[warisa-r]

Here is my attempt to document PERK schemes under the section time integration. Since I think the existing information should preface the brief tutorial of PERK integrator, I put everything in one file but created a different section so that the user can still navigate quickly to the Optimized Scheme section by clicking. However, if you wish to make it a folder and separate the content of this one file into two, being time-integrator-overview.md and optimized-schemes.md, I can also do that.

[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.

[DanielDoehring]

Cool, will take a look next week when I'm back!

[DanielDoehring]

Thanks for starting this! I will use this as a starting point and add stuff to it :)

[DanielDoehring]

@warisa-r I elaborated a bit. Can you go over this and proof-read/check for typos/bad formulations?

[DanielDoehring]

Let' s wait for trixi-framework#2008 then we/you can open the PR at the main repo :)

[DanielDoehring]

Ready to move on to the main repo!