Liu-Zhang cell average limiter (part 2, Euler, CNS)

[jlchan]

Part 2 of #3063. This extends the Liu-Zhang limiter to Euler (and compressible Navier-Stokes, though I have not added any CNS examples since this is already a longer PR).

The main ingredient is project_to_admissible_set for CompressibleEulerEquations1D/2D described in Appendix B and C of https://arxiv.org/pdf/2510.21080. Unfortunately, it is a rather mathematically dense algorithm, and was at least partially translated using AI tools from a reference Fortran implementation. I tried to make it more readable, but the logic is not the easiest to follow.

Additional TODOs:

• Check the case of multiple variables for type stability (since the types of the functions will be different for a tuple like (pressure, density) (see Liu-Zhang global cell average limiter (part 1, advection) #3063 (comment))
• Base.show specialization (moved to Liu-Zhang part 3  #3100)
• try computing the cell averages and the local volumes together (see Liu-Zhang global cell average limiter (part 1, advection) #3063 (comment)). This probably makes more of a difference for P4estMesh, so I think we should defer it for Liu-Zhang part 3  #3100)

[jlchan]

Example of the astro jet with Liu-Zhang: https://gist.github.com/jlchan/37e4b1d5877f3625f63cd55c0968b39d. Without shock capturing, more time-steps are required, but the solution is also less dissipated.

The Liu-Zhang limiter is activated 10 times, with an iteration history of [4, 4, 6, 4, 4, 6, 4, 5, 4, 4]

[jlchan]

Running

include(joinpath(examples_dir(), "tree_1d_dgsem/elixir_euler_leblanc_limiter_liu_zhang.jl"))
using Plots; plot(PlotData1D(sol)["rho"], yaxis=:log)

yields the following:

The limiter is activated 12 times with iteration counts [23, 17, 17, 16, 12, 13, 12, 9, 9, 8, 8, 5]. With only the Zhang-Shu limiter, the elixir crashes about 26% of the way through.

On one thread, this yields the following summary.

─────────────────────────────────────────────────────────────────────────────────────────────
                 Trixi.jl                           Time                    Allocations      
                                           ───────────────────────   ────────────────────────
             Tot / % measured:                  2.34s /  92.6%            204KiB /  49.4%    

Section                            ncalls     time    %tot     avg     alloc    %tot      avg
─────────────────────────────────────────────────────────────────────────────────────────────
rhs!                                23.6k    2.04s   94.4%  86.6μs   3.03KiB    3.0%    0.13B
  volume integral                   23.6k    1.83s   84.6%  77.6μs     0.00B    0.0%    0.00B
  interface flux                    23.6k   69.9ms    3.2%  2.96μs     0.00B    0.0%    0.00B
  Jacobian                          23.6k   37.8ms    1.7%  1.60μs     0.00B    0.0%    0.00B
  surface integral                  23.6k   30.4ms    1.4%  1.29μs     0.00B    0.0%    0.00B
  prolong2interfaces                23.6k   30.0ms    1.4%  1.27μs     0.00B    0.0%    0.00B
  reset ∂u/∂t                       23.6k   28.4ms    1.3%  1.20μs     0.00B    0.0%    0.00B
  ~rhs!~                            23.6k   13.0ms    0.6%   550ns   3.03KiB    3.0%    0.13B
  prolong2boundaries                23.6k    619μs    0.0%  26.3ns     0.00B    0.0%    0.00B
  boundary flux                     23.6k    579μs    0.0%  24.6ns     0.00B    0.0%    0.00B
  source terms                      23.6k    236μs    0.0%  10.0ns     0.00B    0.0%    0.00B
Liu-Zhang positivity limiter        28.3k    100ms    4.6%  3.53μs   1.69KiB    1.7%    0.06B
  Zhang-Shu positivity limiter      28.3k   49.6ms    2.3%  1.75μs     0.00B    0.0%    0.00B
  calc cell averages                28.3k   35.9ms    1.7%  1.27μs     0.00B    0.0%    0.00B
  ~Liu-Zhang positivity limiter~    28.3k   13.0ms    0.6%   460ns   1.25KiB    1.2%    0.05B
  global cell-average limiter          12   1.48ms    0.1%   123μs      448B    0.4%    37.3B
calculate dt                        4.72k   17.7ms    0.8%  3.75μs     0.00B    0.0%    0.00B
analyze solution                        2   3.80ms    0.2%  1.90ms   96.2KiB   95.3%  48.1KiB
─────────────────────────────────────────────────────────────────────────────────────────────