1 - Diffusion
- Use finite difference on \(\displaystyle \frac{\partial f}{\partial t} = \nu\,\triangle f\)
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- Notation: \(f^{t}_{x,y}=f(k_x\,\Delta x,\;k_y\,\Delta y,\;k_t\,\Delta t)\)
- Explicit schemes may oscillates/diverge for large time steps
- \(\Rightarrow\) Use implicit scheme for unconditional stability
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- \(\displaystyle \frac{f^{k+1}_{x,y}-f^{k}_{x,y}}{\Delta t}=\nu \left(\frac{f^{k+1}_{x+1,y}-2\,f^{k+1}_{x,y}+f^{k+1}_{x-1,y}}{(\Delta x)^2} +
\frac{f^{k+1}_{x,y+1}-2\,f^{k+1}_{x,y}+f^{k+1}_{x,y+1}}{(\Delta y)^2}\right)\)
- Assuming \(\Delta x = \Delta y = 1\)
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- \( (1+4\,\nu \Delta t)\,f^{k+1}_{x,y}-\nu\,\Delta t (f^{k+1}_{x+1,y}+f^{k+1}_{x-1,y}+f^{k+1}_{x,y+1}+f^{k+1}_{x,y-1}) = f^{k}_{x,y} \)
- Use Gauss-Seidel iterative method to solve the sparse linear system
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- Initialize \(f^{k+1}=f^{k}\)
- for \(i=1..N_{\max}\)
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- \(f^{k+1}_{x,y} = \frac{1}{1+4\,a}(f^{k}_{x,y}+a(f^{k+1}_{x-1,y}+f^{k+1}_{x+1,y}+f^{k+1}_{x,y-1}+f^{k+1}_{x,y+1}))\) \(\,,\;\;\)
\(a=\nu\,\Delta t\)