Problem

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.)

Methods

1. Fixed-point (FP) iteration with relaxation.
2. Symmetric Gauss-Seidel (SGS) method with SOR.
3. Multicolor Gauss-Seidel (MGS) (if you have a lot of time).

See report for description and results.