# Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $$N+1$$ points in an $$N$$-dimensional space ($$N > 3$$ is possible). At each of these points $$x_i, i=1,...,N+1$$ I know an initial potential/concentration/quantity $$P_i$$. I can evaluate the possible connections between all points. I also have an isotropic diffusion coefficient $$D$$. I am now interested in calculating diffusion between these three points (with mass conservation). I have sketched an example case in 2-D.

This problem sounds rather easy, but I quickly realized that it isn't. Classic finite difference approaches require (to my knowledge) a structured gid, which I don't have. Finite volume could handle unstructured grids, but I cannot afford tessellation in high dimensions (say, $$N > 100$$). For both approaches I would have calculated the first potential derivative along each line (e.g., $$x_1$$-->$$x_2$$), and assumed that the same derivative half a distance outside (dashed lines) is zero. Unfortunately, I don't think this assumption works if I have more than $$N+1$$ points. Finite elements might be an option, but I have yet to find a tutorial which would cover such a case.

Do you have any idea how to solve this system?

Edit: Some additional clarification concerning my goal: I want to simulate the equilibration between the $$N+1$$ vertices. I am only interested in solutions at the vertices themselves, and diffusion would occur along the edges. The specific equation I want to solve is the heat equation:

$$\frac{dP_i}{dt}=K\Delta P_i$$

• Are you interested in computing your quantity at intermediate points? – nicoguaro Mar 31 at 13:06
• No, just at the vertices is fine! – J.Galt Mar 31 at 16:49
• I don't think I'm clear about the exact setup. Where does diffusion act? Along the edges of the polytope described by your $N+1$ vertices? In the interior of the polytope? If it's just along the edges, then you have a bunch of 1d problems coupled at the vertices; since there is an analytic solution for the 1d heat equation, it all reduces to a set of ODEs. In any case, it would be useful to specify which set of equations you want to solve and where. – Wolfgang Bangerth Apr 1 at 23:31
• @WolfgangBangerth, thank you for the comment and sorry for the lack of clarity - I will add some additional information. Yes, I am only interested in diffusion along the edges - I want to simulate the equilibration between these $N+1$ points. The equation I want to solve is the heat equation at the vertices, which requires second derivatives. The approach you suggest sounds intriguing - could you give me a quick hint on how to approach this solution? – J.Galt Apr 2 at 9:48
• I think I still don't understand. You say $P_i$ is the potential at vertex $i$. That's a time-dependent function. Then how do you define $\Delta P_i$? $\Delta$ is a spatial operator, but $P_i$ does not depend on any spatial variables. – Wolfgang Bangerth Apr 2 at 14:14