Caching uses too much memory

Nobody has claimed this yet.

In `TrixiShallowWater.jl` I noticed error messages from `julia-actions/cache@v1` that old caches could not be deleted. This seems to be solved by setting some permissions explicitly as suggested in https://github.com/julia-actions/cache.

I was wondering if this should be also done in `Trixi.jl` or why this is not frequently encountered in `Trixi.jl`. Looking at the CI runs there I saw that this happens rarely as [here](https://github.com/trixi-framework/Trixi.jl/actions/runs/7732344766/job/21082071822#step:23:7), but most of the times it says "No existing caches found".

The reason for this is probably that our caches get deleted by github because we use too much memory. Looking at the current cache status, a single test job saves a cache of around 1GB. If I am correct it means that even a single CI run creates around 20GB of caches, so twice our cache limit.

Github will then start to delete old caches until it reaches the 10GB limit. Besides that, from what I have seen all CI jobs can still load a cache (probably also because the limit is not enforced so strictly), so it doesn't seem to cause problems right now.

I still think it would be good to keep an eye on this and find some strategies to reduce our generated cache data.

Start by inspecting the CI cache configuration and cache-saving steps associated with the linked GitHub Actions run, then compare them with julia-actions/cache@v1 and the TrixiShallowWater.jl setup. Measure how many caches are produced and their approximate sizes. Done requires an agreed strategy that reduces generated cache data or explains why the current behavior is acceptable.

Written by the indexing model from the issue text.