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35 lines
1.3 KiB
Org Mode
35 lines
1.3 KiB
Org Mode
#+TITLE: Diffusion module
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This is the according repository to the diffusion module we discussed earlier.
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With this readme I will document all my steps I've done and will do.
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* Theory
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- $\alpha$ - diffusion coefficient (dependent on species and direction(?))
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- $h=1/M$ : with $M^2 = [0,1]^2$ - grid divided into parts between 0 and 1
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(/spatial step/)
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- $k=T/N$ : with $N = [0,T]$ - time step size
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- coefficients of the given equation from the paper are:
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- $\alpha_xk/h^2$ in x direction
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- $\alpha_yk/h^2$ in y direction
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- $1+2*(\alpha_xk/h^2) + 2*(\alpha_xk/h^2)$ for the same grid cell with n+1
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time step
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So as a conclusion: We get a system of equations to solve for $u$. Maybe use
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LU-Decomposition here. It is easy to implement, deterministic and also
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performant. Since each $u_j$ is dependent on $u_{j-1}$ this will be hard to
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parallelize but I will keep parallelization in mind.
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Regarding the borders: I'm not quite sure what to do. Maybe it might be a good
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idea to use a simple gaussian kernel here to smooth those two columns and two
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lines.
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* Implementation
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So currently I consider to implement the following methods for the module:
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- +decompose matrix A into L and U+
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- better use a library like Eigen here:
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- using =SparseMatrix= to represent matrix $A$
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- =SparseLU= to solve
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