Update Documentation: Optimized Schemes, PERK Section, and PairedExplicitRK2 Tutorial

[warisa-r]

Following the merge of #2008 and per @DanielDoehring's instructions, this pull request updates the documentation, especially in the file: time_integration.md to capture recent changes and improvements:

• Added a description of optimized explicit schemes
• Provided details on the optimization problem for maximizing timestep stability
• Included references to key publications and methods for solving the optimization
• Added a brief tutorial on using the PairedExplicitRK2 time integrator

Please let me know if you'd like any changes and improvements!

Xref: #21

[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]

Did you check how the rendered version looks like?

Yeah, the locally rendered version of time_integration.md looks like this:

.

I just saw that the numbers in front of the step instructions here are all 1, the same as the numbering in my locally rendered version. However, in the time_integration.md file, every step is numbered correctly as in 1., 2. 3.. Could you help instruct how to fix this?

[JoshuaLampert]

I just saw that the numbers in front of the step instructions here are all 1, the same as the numbering in my locally rendered version. However, in the time_integration.md file, every step is numbered correctly as in 1., 2. 3.. Could you help instruct how to fix this?

Seems like a markdown thing. Can you try to indent the code blocks in each of steps by four spaces?

[warisa-r]

I just saw that the numbers in front of the step instructions here are all 1, the same as the numbering in my locally rendered version. However, in the time_integration.md file, every step is numbered correctly as in 1., 2. 3.. Could you help instruct how to fix this?

Seems like a markdown thing. Can you try to indent the code blocks in each of steps by four spaces?

I think this might solve the problem but when I added the indentation, an error that I have been trying to fix shows up. I rechecked that all code blocks have consistent indentations but it still shows up:

┌ Error: failed to parse exception in src\time_integration.md
│   exception =
│    ParseError:
│    # Error @ none:15:8
│ 
│        
@example PERK-example-1
│    #      └─────────────────────┘ ── extra tokens after end of expression
└ @ Documenter C:\Users\user\.julia\packages\Documenter\Bs999\src\utilities\utilities.jl:44
ERROR: LoadError: AssertionError: Issue at #= time_integration.md:138 =#:

[warisa-r]

I don't think we need the numbered list to show the steps the user needs to take if they want to use the integrator. If we need to explicitly name any step of the tutorial, we can simply just use a fifth-level heading. Let me know if you want anything else changed!

[ranocha]

Fine from my side if @DanielDoehring approves and the docs rendering has been checked.

[ranocha]

Is this ready to be merged, @DanielDoehring?

[DanielDoehring]

Yes, good to go! Thanks @warisa-r and @JoshuaLampert, @ranocha !