# Numerically solving unbounded Fokker-Planck equation

I am wanting to solve a 3D Fokker-Planck equation of the form:

$$\partial_{t}p(\mathbf{x}, t) = -\nabla \cdot \mathbf{J}$$

where $$\mathbf{J} = \mathbf{v}(\mathbf{x})p(\mathbf{x}, t) - D\nabla p(\mathbf{x}, t)$$ with $$p(\mathbf{x}, t)$$ the probability distribution as a function of space and time, $$D$$ is the diffusion constant and $$v(\mathbf{x})$$ is the constant velocity vector for a flow.

I have incorporated a finite difference numerical scheme with reflective boundary conditions in the 2D plane, but now would like to incorporate infinite boundary conditions in the $$\pm z$$ directions. For accurate results this requires a very large mesh size, and thus takes too long.

The question is whether I would be able to use a finite difference scheme for the $$x$$-$$y$$ direction combined with a spectral method using Hermite functions for dealing with the $$z$$-direction as in the link: https://core.ac.uk/download/pdf/56367916.pdf to do this (firstly for the more simple case of $$\mathbf{v}(\mathbf{x}) = \mathbf{v}(x, y)$$ such that there is only diffusion in the $$z$$-direction)

• Is there a reason you think you couldn't do that? For what it's worth, spectral methods in some directions and finite <something> methods in the remaining directions are usually called pseudo-spectral methods. – tpg2114 Aug 8 '19 at 13:22
• Yes, I believe it would be possible, but I am not sure about the full details. My reasoning is : if $p(\boldsymbol{x}, t) = \sum f_{n}(x, y, t)\frac{1}{\sqrt{2^{n}n!}} H_{n}(\alpha z)e^{-\alpha^{2}z^2}$, then 1. find $f_{n}(x, y, t = 0)$ from initial distribution. Substitute the expansion into the FP equation to get some relation between $f_{n}$ and derivatives of $f_{n}$ (then approximated using finite difference). 2. Time step. 3. Sub new $f_{n}$ into the expansion to get $p(x, y, z, t+\Delta t)$? And i believe using this method, a mesh in the z-direction is not required. Is this correct? – Hello Aug 8 '19 at 13:28
• By no mesh in z-direction I mean, say I would like to only know at all times $p(x, y, z=0, t)$ then I only need to effectively do the above procedure once and insert $H_{n}(0)\times e^{-\alpha^{2}\times 0}$ in the above expression – Hello Aug 8 '19 at 13:34
• Check this post: nicoguaro.github.io/posts/hermite_ritz_qm – nicoguaro Aug 8 '19 at 15:50