# Grids for atmosphere simulation with finite volumes on the globe

I am currently in the early construction process of building a simple CFD model of a rotating planetary atmosphere. The planet should be allowed to tilt significantly, so that a time-dependent source function for heating the atmosphere can also go overhead the poles.

Because of this, I have to consider grids other than the popular longitude-latitude grid in atmospheric (earth) science. My first instinctive move was to look for a tesselation of the sphere which is more agnostic towards the planetary rotation.
I found a classic Sadourny paper that also provides a simple algorithm how to generate said tesselation using triangles. I implemented this and it works nicely to generate grids of arbitrary refinement.

However, it turns out (Staniforth, grid review) that triangles are bad and one should use quadrilaterals instead. A quadrilateral tesselation based on a icosahedron as lowest resolution grid is still possible.

Now the latter review I've mentioned did comment on grid problems using finite difference methods, which i do not intend to use. My first question would now be:

• Does a triangular tesselation of spheres with finite volume methods have the same problems as finite difference methods with noise and accuracy? Also I imagine, computing a FV method like the WAF (weighted averaged flux, mentioned in the Toro book, chapt. 16 to be easily implementable for complex geometries??) on triangles to be awkward, because of the geometrical enlargment of a wave into the next triangle.

and because of those considerations (on the risk that that's probably a more opinion-based question)

• Would numerics on a sphere simplify if I just abandon the triangular tesselation and go for a quadrilateral one, by pairwise joining triangles? Is there maybe a similar review like the Staniforth (see link above) just for finite volume methods?
• What about using a "cubed sphere"? As presented in Figure 5 in your second reference. – nicoguaro Aug 30 '19 at 16:12
• There are so many different ways to grid a sphere that it's hard to suggest one. @nicoguaro suggested the cubed sphere is very popular. There's also the "pillow sphere" and the MPASS grid structure. – Kyle Mandli Aug 30 '19 at 21:11
• @KyleMandli I know, that's why I restricted my research, programing and questioning to the triangilar tesselation. I guess the point of my question is "are triangles + FV worth it, or should I directly abandon them and go for quadrilaterals.."? – AtmosphericPrisonEscape Aug 31 '19 at 0:39
• Triangles are not inherently bad, but they are only worth the trouble if you have to deal with very complex geometry. There isn't anything much simpler than a sphere, and I would add my voice to advocates of the cubed sphere. You still have the options to combine this with FE/FD/FV. – Philip Roe Aug 31 '19 at 4:21
• @AtmosphericPrisonEscape I am starting to work on icosahedrons now thanks to ICON. Did you get this going? – gansub Apr 1 at 6:06