Thanks for raising this @valeriabarra. Are you referring to the appendix of Guba et al? As I read it, they're describing the same thing we do, but writing it in component form. Note that for scalar problems, we don't use an inscribed cube at al. In the body, I see they discuss the hyperviscosity operator, but not advection.

This might be okay for explicit models, but implicit would like good conditioning with respect to the global dofs of the system, and having the polar singularity (which they imply is benign non-uniqueness here) is a real issue for implicit.

Thanks for your reply.

I was mainly referring to the transformation of coordinates with local systems on each panel. So yes, their approach described in Appendix A that is arbitrary-quadrilateral-based instead of panel-based would eliminate the need for a lot of auxiliary work that was needed in the panel-based approach (see for instance these FindPanelEdgeNodes and TransformCoords functions that I had worked on). The local maps defined on each element would eliminate the need for if/else or switch branching because they do not need to know if you are transforming coordinates from say panel A to panel B.

Unless I'm unable to read, the appendix only addresses how to differentiate on the surface of the sphere, not how to represent vectors. We derived the same formulation for our BPs on the sphere. But for shallow water, we need to represent tangential vector fields in a basis that doesn't have a singularity and in which we can process all nodes of an element in a consistent chart.

One way to do that (nodally) is to take a Householder QR factorization of the one-column matrix consisting of the radius vector. The last two columns of Q are orthogonal and the process to compute them (a single Householder reflection) is backward stable. But one still needs to convert from a separate chart per node to a single chart per element. Note that this is only an issue for representing tangential vectors.

Of course you can read! I was just referring to one sentence in the intro of the Appendix.
Yes, we derived the same formulations to differentiate on the surface of the sphere for the BPs :)
I was just pointing out another piece of literature that I had discovered where this was formalized. But yes, in particular Guba et al. only briefly mention the treatment of vector fields in the text (see beginning of Sec. 2.4).

Anyways, if anyone in the group would like to discuss any of this further or would like to help/collaborate/take over, just let me know!

Hi all!
I think it's really ok to close this issue and its related PR - It can always be restored later if anyone wants to pick up that work. My apologies. Thanks.