# Numerical scheme for the heat equation on the icosahedral hexagonal grid

I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity of a grid, but I did not find a way to port this method on a sphere.

I want to solve the heat equation on such a grid:

$$\frac{\partial u}{\partial t} - \Delta u = f(x, y, t)$$

with some simple border conditions for $$u$$

I failed to find good resources with numerical schemes and proof of their stability. There is an obvious scheme:

Hexagon:

$$\frac{u_i^{n + 1} - u_i^n}{\Delta t} = \frac{1}{6}\cdot \frac{ - 6u_i^n + \sum\limits^{6}_{j = 1}u_j}{d^2} + f^n_i$$

Pentagon:

$$\frac{u_i^{n + 1} - u_i^n}{\Delta t} = \frac{1}{5}\cdot \frac{ - 5u_i^n + \sum\limits^{5}_{j = 1}u_j}{d^2} + f^n_i$$

where $$u_j$$ are neighbours of $$u_i$$ and $$d$$ is a half of center-to-center distnance.

This scheme's error is probably $$O(\Delta t) + O(\Delta h^2)$$ as it is proofed here behind a paywall for a regular hexagonal grid, not for spherical one with the pentagon though. I bet there are some instabilities that are occurring near pentagons. Maybe they will go away by making the scheme implicit, I don't know.

Another problem is how to store such a grid, I know there is a way to store such a grid if it is on a plane, but I could not port it to a sphere on my own.

And another question that I have is how to extend the hexagonal grid from only the surface to some depth, making it model a spherical layer.

There are some resources for a hexagonal grid on a plane, but it seems like nothing for one on a sphere.

To summarise:

• Where to find materials to get good schemas for the heat equation on this grid?
• How to store it?
• How to extend it to model spherical layer?
• It would probably be easier if you add the centerpoints and subdivide the polygons into triangles using radial segments. Then you can use standard finite-element approaches to a triangular grid. Apr 18, 2022 at 11:55
• That is, sadly, true. There are much more resources for triangular meshes than for this one. If you got some good papers and/or tutorials though, I would appreciate them. Also, is there any better strategy for combining triangles' values than to average them? Apr 18, 2022 at 12:47
• I second the suggestion of converting the mesh to a triangular one. Nothing is to be gained from sticking with non-standard element shapes. Apr 18, 2022 at 17:20
• Stability and accuracy both have to do with a limit in which the grid gets small. But small compared to what? In 2d small compared to a typical feature of the solution. Here, also small with respect to the curvature of the sphere. Apr 21, 2022 at 20:12