MPI: Barriers, intermediate output, and debugging

> It was used to ensure that when root reports that the initial conditions have been set, all ranks have reached this point.
> What could be a good way to replicate such a behavior in Taal? If we don't have a good answer right now, we should create an issue for it, since this is something I have relied on frequently in the past to debug MPI runs.

Xref https://github.com/trixi-framework/Trixi.jl/pull/292#discussion_r519791509

There are some comments containing the link (https://github.com/trixi-framework/Trixi.jl/issues/328) to this issue at some relevant places in the source code.

Start with the discussion in Trixi.jl#292 and inspect the source-code locations that link to issue #328. Clarify the desired barrier and intermediate-output behavior for debugging MPI runs, then determine what a complete solution should demonstrate across ranks.

Written by the indexing model from the issue text.