From e136e0019245cfc4d59d2a2b2e4323fd20a1f39a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Max=20L=C3=BCbke?= <mluebke@gfz-potsdam.de>
Date: Wed, 13 Oct 2021 10:15:44 +0200
Subject: [PATCH] update README file

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+#+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 steps per iteration (/time step/)
+- 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.