Positivity violation in AMR

[Arpit-Babbar]

Zhang and Shu's scaling limiter uses admissibility of the means to enforce admissibility of the solution at the solution points. Quoting the remark from our preprint,

the procedure (Zhang and Shu's scaling) doesn’t ensure that the polynomial is admissible at points which are not the solution points. Adaptive mesh refinement introduces such points into the numerical method and can thus cause a failure of admissibility preservation in the following situations: (a) mortar solution values obtained by interpolation are not admissible, (b) mean values of the solution values obtained by interpolating from the larger element as in are not admissible.

I have not run into this issue in Trixi.jl, but I did when I was using Trixi's positivity limiter and AMR callbacks in TrixiLW.jl. Then, in our work, we did the natural fix as follows

Since the scaling limiter can be used to enforce admissibility of solution at any desired points, the remedy to both the issues is further scaling; we simply perform scaling of mortar solution values, interpolated solution point values with the admissible mean value of the larger element. This will ensure that the mortar solution point values and the mean values are admissible.

To fix this, we need to begin with #215.