Boundary gradients for quadrilaterals in Green-Gauss

[bigfooted]

From Nishikawa: https://www.researchgate.net/publication/266446943_Beyond_Interface_Gradient_A_General_Principle_for_Constructing_Diffusion_Schemes

At boundaries, at the moment we only do B.29, so the boundary gradients for quadrilateral cells are only correct for parallelograms. We actually need B.28.
We use the dual approach where we connect edge centers to obtain the cell centroid. But for the alternative dual, where we simply take the geometric mean as cell center, the last two terms of B.28 drop out. I see some ways to improve boundary gradients:

1. For quadrilaterals, we could use the geometric mean and keep the rest the same. The quality of the boundary gradient would improve because instead of missing lines 2,3, and 4 of B.28, we are only missing line 2.
2. We could additionally compute the terms in line 2. This might mean we need some additional information on the right and left normals.
   Any thoughts? Would this be worth correcting?

[pcarruscag]

We may as well change SU2 to be cell-centered if we are going to try accounting for all that detail.
Green-Gauss gradients are innacurate in most situations, but they are robust, and consistent with how fluxes are integrated, which is a requirement for dealing with some source terms (force balanced discretizations).
If we need more accurate gradients we should look for other methods.

[bigfooted]

OK, so the design guideline should be 'implement improvements that have a good improvement/complexity ratio'.
I'll check if there is something in the paper that's worth investing some time in...

Green-Gauss gradients are innacurate in most situations

Well, that's because we don't have all these fancy corrections, aye? ;-)

[pcarruscag]

They also become inaccurate in cell-centered meshes, there are many proofs showing they become 0th order.
By the time one fixes the accuracy problems the method is either too costly, too unstable, or both.