Asymptotic Telegrapher’s Equation (P₁) Approximation for the Transport Equation

Abstract

The diffusion approximation for the Boltzmann (transport) equation suffers from several disadvantages. First, the diffusion approximation succeeds in describing the particle density only if it is isotropic, or close to isotropic. This feature causes the diffusion approximation to be quite accurate for highly isotropically scattering media but to yield poor agreement with the exact solution for the particle density in the case of nonisotropic behavior. To handle general media, the asymptotic diffusion approximation was first developed in the 1950s. The second disadvantage is that 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 P₁ approximation (which gives rise to the Telegrapher’s equation) has a finite particle velocity but with the wrong value, namely, υ/√3. In this paper 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.