You could do SWE with finite elements, but generally finite volumes would be your choice to handle any potential discontinuities and is more stable and accurate for practical problems.

Here is a tutorial. https://www.tfd.chalmers.se/~hani/kurser/OS_CFD_2010/johanPi...

What's unclear to me is how do I model the spherical geometry without exploding the complexity of the solution. I know that a fully custom mesh with a pile of formulas for something like beltrami-laplace operator would work, but I want something more elegant than this. For a example, can I use the Fibbonacci spiral to generate a uniform spherical mesh, and then somehow compute gradients and the laplacian?

I suspect that the stability of FE or FV methods is rooted in the fact that the FE functions slightly overlap, so computing the next step is a lot like using an implicit FD scheme, or better, a variation of the compact FD scheme. However I'm interested in how an adept in the field would solve this problem in practice. Again, I'm aware that there are methods of solving such systems (Jacobi, etc.), but those make the solution 10x more complex, buggier and slower.

Unless "normally" you mean the normal distribution, which indeed has zero skewness.

Basic politeness is absolutely dead, nobody has any concept of acknowledging they are asking for a favour; we just blast Instagram/TikTok reels at top volume and smoke next to children and elderly in packed public spaces etc. I'm 100% sure it's not rose-tinted memories of the 90s making me think, it wasn't always like this...

Typically, Finite Volume Method is used for fluid flow problems. It is possible to use Finite Element Methods, but it is rare.