PDE solver on 2D sphere

Hi,

I would like to solve a spatio-temporal non-linear PDE on a 2D sphere (not a 'shell' with a finite thickness).

I see there is a number of mesh generators (cubed sphere, icosahedral grid) for Matlab, but how to integrate them with a PDE solver?

Is it possible with PDEtoolbox or other Matlab tools? Any examples?

Climate modelers probably do this all the time...

[Amal George M]

Hi Mark,

MATLAB Partial Differential Equations Toolbox support both 2D and 3D geometries. You can find a couple of examples at this link . Time dependency can also be integrated into the problem, by providing a list of time instances to the ' solvepde ' function.

Syntax:

result = solvepde(model,timelist);

Here is an example involving time-varying results.

Hope this helps.

[Mark vanRossum]

Thank you, but I would like to solve on a sphere, which is a 2D object embedded in 3D space.

I don't see how that fits in the PDEtoolbox which either requires 3D objects (which a sphere is not) or 2D objects embedded in 2D space.

The only solution I currently see is to make it a shell of finite thickness.

[Torsten]

You could calculate on a rectangle if you use spherical coordinates ...

Best wishes

Torsten.

[Mark vanRossum]

That's a good idea Torsten. Will the poles give trouble, and do I need special boundary conditions there?

[Torsten]

I don't know. My guess is that the periodic boundary condition will be the most difficult part to deal with when using the PDE toolbox.

Best wishes

Torsten.

Answer 1

[Ravi Kumar]

If you are referring to thin walled hollow sphere, a 3-D volume as your problem domain, then you can create such geometry in PDE Toolbox as:

radius = 1;
thickness = radius/10;
gm = multisphere([radius-thickness, radius], 'Void',[1,0]);
model = createpde;
model.Geometry = gm;
pdegplot(model,'FaceLabels','on','FaceAlpha',0.3,'CellLabels','on')
print('Hollow_Sphere','-dpng','-r200')