Let's assume I am working on a 2-D domain on $R^2$, with my coordinates $x \in[-1,1]$, $y \in[-1,1]$ and I want to solve a popular CFD problem, like the shallow water system or the Euler system.
At $x=0$ there is some physical feature that I want to resolve really well, so I want to employ mesh refinement around the origin. I am restricted to squares as resolution element.

I know that my solution around the origin will be close to hydrostatic, but possibly evolving and rotating slowly. This could be e.g. a cyclonic system.

I would now be interested in the following:

Which static technique of mesh refinement, continuous (e.g. with $dx \propto 1-exp(-x^2)$) or step-wise (often also called nested meshes) would give me the smaller errors in my solution?

I imagine that this may have been done already in the atmospheric community, which uses global grids with local refinement, but so far I came up with only step-wise refinement techniques. Also this review paper on cubed sphere grids mentions that non-uniformity is bad for a dynamic situation, but that's it.

Any answer that would also give a citable (review) paper to answer this question would be greatly appreciated.


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