Numerically solving a PDE on an unit tangent bundle

Question

Let $M$ be a manifold and $UT(M)$ its unit tangent bundle. I have a PDE which looks something like $$ \frac{\partial f}{\partial t}(x,v) = (v\cdot \nabla_x) f(x,v) + \Delta f(x,v) $$ where $x\in M$, $v \in T_xM$, i.e. $x$ is a point on the surface and $v$ is an unit vector at that point, $\nabla_x$ is gradient on the surface $M$ and $\Delta$ is laplacian on the unit tangent bundle.

Is there any software which would allow me to solve such equation numerically?

My main concern is how do I obtain discretization of the unit tangent bundle? I would like to have a triangular mesh of my surface and then somehow get a discretization of its unit tangent bundle. Once I have the mesh I can probably use some standard library to solve the PDE.

Answer 1

The equation you are trying to solve in in essence the analogue of the "radiative transfer" equation when posed on a manifold. There is a vast literature in this area -- both in astrophysics and in nuclear physics, two areas in which this equation is important --, with the obvious difficulty being that your domain is pretty high-dimensional: if your manifold is three-dimensional, then you have a five-dimensional problem. (For 2d manifolds, the problem is three-dimensional.)

Typical ways to address this are:

• To approximate the equation by a sequence of equations that resemble diffusion equations in $x$
• To discretize the angular direction $v$ on a separate mesh from the spatial mesh, and then obtain a tensor product mesh in both $x$ and $v$.

There are many other approaches, such as ray tracing or Monte Carlo methods. Search the literature!

Answer 2

I looked around a bit and found this paper (featuring no less than Shing-Tung Yau!) about the problem of generating meshes for the unit tangent bundle $UT(S^2)$ on the 2-sphere $S^2$. As you point out, $UT(M)$ probably isn't diffeomorphic to $M \times S^n$. For example in Yin's paper they point out that $UT(S^2)$ is non-trivial and can't be embedded in $\mathbb{R}^3$. However, locally the bundle is trivializable, and Yin's paper suggests that one approach could be to divide the manifold into trivializable patches, then explicitly compute the transition maps between patches.

You might be able to implement their idea to generate turn a mesh for $M$ into a mesh for $UT(M)$, which would have to be embedded into a much higher-dimensional Euclidean space. While many finite element libraries (deal.II among them) can solve PDEs on surfaces, I think they usually assume that it's a 1D or 2D surface embedded in 2D or 3D respectively. If you were working with something as simple as $UT(S^2)$, you'd have to embed it in $\mathbb{R}^4$, which probably falls outside the purview of many libraries.

You could also try to work only with a mesh for $M$ or, more in the spirit of the paper, meshes for a set of overlapping patches $M_\alpha$ such that $UT(M)$ is trivializable on each $M_\alpha$. Provided you have a way to calculate the transition maps, you can represent $f$ within each patch as a tensor product of a Galerkin basis function $\phi$ defined on $M_\alpha$ with the spherical harmonics of the appropriate dimension. You could then try and solve your PDE using, say, the Schwarz alternating method or some other domain decomposition method. Of course you'll also have to partition the mesh first. METIS is a great tool for this, but you could very well get a partition with a patch that isn't trivializable. It might be possible to then recursively partition non-trivial patches. Or it might be easiest to implement this yourself.