Solve simplied Radiative Transfer Equation (RTE) with 6 directions with scattering only.
$$\begin{align*}
\frac{\partial F_1}{\partial x} &= \sigma (\frac{1}{6} \Sigma F_i - F_1) \\[1em]
-\frac{\partial F_2}{\partial x} &= \sigma (\frac{1}{6} \Sigma F_i - F_2) \\[1em]
\frac{\partial F_3}{\partial y} &= \sigma (\frac{1}{6} \Sigma F_i - F_3) \\[1em]
-\frac{\partial F_4}{\partial y} &= \sigma (\frac{1}{6} \Sigma F_i - F_4) \\[1em]
\frac{\partial F_5}{\partial z} &= \sigma (\frac{1}{6} \Sigma F_i - F_5) \\[1em]
-\frac{\partial F_6}{\partial z} &= \sigma (\frac{1}{6} \Sigma F_i - F_6)
\end{align*}$$
With boundary conditions
$$\begin{align*}
F_1(-1, y, z) &= F_b(y, z) \\\
F_3(x, -1, z) &= F_b(x, z) \\\
F_5(x, y, -1) &= F_b(x, y) \\\
F_2(1, y, z) &= 0 \\\
F_4(x, 1, z) &= 0 \\\
F_6(x, y, 1) &= 0
\end{align*}$$
(If any of the terms related to the physics is incorrect, please forgive me.)
- Fixed-point (FP) iteration with relaxation.
- Symmetric Gauss-Seidel (SGS) method with SOR.
- Multicolor Gauss-Seidel (MGS) (if you have a lot of time).
See report for description and results.