Time domain RTE

The time dependent radiative transport equation is stated as

$$\begin{array}{c}\begin{array}{rl}& \frac{1}{c}\frac{\partial u}{\partial t}+\hat{s}\cdot \nabla u=\sigma (B(\nu ,T)-u)\\ & \\ & C_{\nu }\frac{\partial T}{\partial t}={\int }_{S^{d-1}}{\int }_0^{\infty }\sigma (u-B(\nu ,T))d\nu d\hat{s}\end{array}\end{array}$$

which describes the interaction of material and radiation. where $u(x,t,\hat{s},\nu )$ is radiation intensity, $T=T(x,t)$ is material temperature. $\sigma =\sigma (x,\nu ,T)$ is opacity thickness, $C_{\nu }$ is heat capacity. $B(\nu ,T)=\frac{2h}{c^3}\frac{{\nu }^3}{\exp (h\nu /kT)-1}$

by using approximation, the equation can be rewritten as an easier one,

$$\begin{array}{c}\begin{array}{rl}& \frac{1}{c}\frac{\partial u}{\partial t}+\hat{s}\cdot \nabla u+\sigma u=\frac{\sigma b}{|S^{d-1}|}ac{T}^4\\ & \\ & C_{\nu }\frac{\partial T}{\partial t}={\int }_{S^{d-1}}{\int }_0^{\infty }\sigma d\nu {\int }_{S^{d-1}}u\hat{s}-{\sigma }_{p}ac{T}^4\end{array}\end{array}$$

where $b$ is average of $B$, ${\sigma }_p$ is average of $\sigma b$.

There is a way to decouple the equations, by taking $t$ as another spatial variable. $z=(x,t)$, we have

$$\tilde{s}\cdot {\nabla }_{z}u+\sigma (z,\nu )u=H(z,\nu )$$

which is quite easy to come up with a solution for $\varphi ={\int }_{S^{d-1}}u$. And insert into temperature equation.

$C_{\nu }\frac{\partial T}{\partial t}={\int }_0^{\infty }\sigma d\nu {\int }_{S^{d-1}}{\int }_0^{{\tau }^{-}(z,\tilde{s})}\exp (-{\int }_0^p\sigma (z-\mu \tilde{s})d\mu )H(z-p\tilde{s})dpd\tilde{s}-{\sigma }_{p}ac{T}^4$

And solving this can apply some fast algorithm like treecode or FMM as we did before, forward Euler scheme will generate $O(\Delta x)$ error.