solve 2D and 3D Poisson equations on the HEALpix grid using xdggs capability?

[tinaok]

Hi, I had question from @shokun5656 during The World Climate Research Programme Global KM-scale Hackathon

I want to solve 2D and 3D Poisson equations on the HEALpix grid system. I believe that the multigrid method, which is known as a fast Poisson solver, would work well with the HEALpix grid system. Does such a solver already exist? If not, is there a demand for such a solver in the community?

@jmdelouis is there any example from your domain of science?

If not existing, can xdggs and associated tooling can help solving this problem? or any extension of tool necessary?

From what I understood it requires at least

• cell center coordinates: Dataset.dggs.cell_centers()
• cell boundary polygons: Dataset.dggs.cell_boundaries()
• N+1 neighbours: https://github.com/IAOCEA/healpix-convolution/blob/main/experimentation/neighbours.ipynb?
• N+2 neighbours: https://github.com/IAOCEA/healpix-convolution/blob/main/experimentation/neighbours.ipynb?

Does anyone have thoughts on this?

[jmdelouis]

It is already handled by FOSCAT. The Poisson equation is equivalent to computing the Laplacian. The sum of the real parts of the oriented wavelets provides an estimate of the Laplacian. Therefore, FOSCAT can compute the Laplacian in a single line of Python, and the Poisson equation can be solved in two ways:

• by gradient descent, since FOSCAT supports it, or
• by conjugate gradient inversion, which is likely better and faster in this specific case.

If you have any data I can release a notebook that would show how to call FOSCAT to solve the Poisson equation.

A futur fonction can be easily added to foscat that computes the Laplacian of a 2D and 3D fields using Healpix grid.