Non-Uniform WENO Core (Direct Derivative Reconstruction)

[utkuyilmaz1903]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[utkuyilmaz1903]

Hi @ChrisRackauckas @xtalax,

Briefly, _weno_f_nonuniform_core implements unrolled Fornberg weights with Shi-Hu-Shu. It is strictly zero-allocation and verified against ForwardDiff/Symbolics.

Since it's currently a node-centered, central, non-conservative core, I’d love your high-level sanity check on two quick points before I start coding the boundary cases:

Boundaries: I plan to handle edge nodes via biased Fornberg stencils (dropping to 3rd-order locally, which should maintain global 4th-order per Gustafsson’s GKS theory). Does this sound like the right path?

Upwinding: Since the current uniform weno_f also behaves centrally, should we keep this strictly as a central operator for now, or eventually discuss a higher-level wrapper that passes advection velocity?

No rush for a detailed line-by-line review just wanted to make sure the foundation is on the right track before spending time on the edge dispatches.

[ChrisRackauckas]

all includes should be at the top level.

[ChrisRackauckas]

This is already implemented in the package?

[utkuyilmaz1903]

I actually considered using it first, but I noticed it allocates dynamic arrays internally, which causes heap allocations.I felt using the explicit algebraic form here was the safest bet to guarantee zero allocation.

[ChrisRackauckas]

Looks good once it's out of draft.