#+TITLE: Diffusion module

This is the according repository to the diffusion module we discussed earlier.
With this readme I will document all my steps I've done and will do.

* Theory

- $\alpha$ - diffusion coefficient (dependent on species and direction(?))
- $h=1/M$ : with $M^2 = [0,1]^2$ - grid divided into parts between 0 and 1
  (/spatial step/)
- $k=T/N$ : with $N = [0,T]$ - time step size
- coefficients of the given equation from the paper are:
  - $\alpha_xk/h^2$ in x direction
  - $\alpha_yk/h^2$ in y direction
  - $1+2*(\alpha_xk/h^2) + 2*(\alpha_xk/h^2)$ for the same grid cell with n+1
    time step

So as a conclusion: We get a system of equations to solve for $u$. Maybe use
LU-Decomposition here. It is easy to implement, deterministic and also
performant. Since each $u_j$ is dependent on $u_{j-1}$ this will be hard to
parallelize but I will keep parallelization in mind.

Regarding the borders: I'm not quite sure what to do. Maybe it might be a good
idea to use a simple gaussian kernel here to smooth those two columns and two
lines.

* Implementation

So currently I consider to implement the following methods for the module:

- +decompose matrix A into L and U+
- better use a library like Eigen here:
  - using =SparseMatrix= to represent matrix $A$
  - =SparseLU= to solve