1.3 KiB
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
SparseMatrixto represent matrix $A$ SparseLUto solve
- using