Deriving a modified asymptotic telegrapher's equation (P₁) approximation

The well known asymptotic diffusion approximation was first developed in the 50's by Frankel and Nelson, and expanded by Case et al. and by Davison, to handle the asymptotic steady-state behavior. But, in time-dependent problems, the parabolic nature of the diffusion equation predicts that particles will have an infinite velocity; particles at the tail of the distribution function will show up at infinite distance from a source in time t = 0+. The classical P ₁ approximation (or the equivalent Telegrapher's equation) has a finite particle velocity, but with the wrong value, namely υ/√3. In this work we develop a new approximation from the asymptotic solution of the time-dependent Boltzmann equation, which includes the correct eigenvalue of the asymptotic diffusion approximation and the (almost) correct time behavior (such as the particle velocity), for a general medium. The resulting scalar flux from the new approximation shows a good agreement with the exact solution of the Boltzmann equation.