I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation.
Consider the following penalized transport equation:
$$\begin{cases} \lambda f_\lambda + \sin\phi\frac{\partial f_\lambda}{\partial\eta} + F(\eta)\cos\phi\frac{\partial f_\lambda}{\partial\phi} + f_\lambda = H(\eta,\phi)\\ f_\lambda(0,\phi) = h(\phi) & \text{ for }\sin\phi < 0\\ f_\lambda(L,\phi) = f_\lambda(L,-\phi). \end{cases}$$
Assume everything is as nice as necessary; in particular, $h$ and $H$ are bounded. Here $\eta\in[0, L]$ and $\phi\in S^1$. Now let $V(\eta)$ be a function such that $F(\eta) = -\partial_\eta V(\eta)$.
Let $\mathbf{x}(s) = (\phi(s),\eta(s))$ be the characteristic curve corresponding to the PDE. Using the method of characteristics, we deduce that the characteristic curve $\mathbf{x}(s)$ satisfies $\mathbf{\dot{x}}(s) = (\phi'(s),\eta'(s))= (F(\eta(s)\cos\phi(s), \sin\phi(s))$. Simple computation shows that along this curve, the quantity $\cos(\phi(s))e^{-V(\eta(s))} \equiv E$ remains constant. This proves (3.42) in the paper.
The authors then claim that along such a curve, the equation can be simplified as follows: $$\lambda f_\lambda + \sin\theta\frac{\partial f_\lambda}{\partial \eta} + f_\lambda = H.$$ It's not clear to me how this follows. Indeed, along the characteristics, we have $\cos\phi = Ee^{-V(\eta(s))}$, but plugging this back into the original PDE doesn't get rid of the $\frac{\partial f_\lambda}{\partial\phi}$ term.