Deriving the Time-Dependent Asymptotic PN Approximation

  Re'em Harel  ,  Re'em Harel [1]  ,  Stanislav Burov [1]  ,  Shay I. Heizler [2]  
[1] Department of Physics, Bar-Ilan University, IL52900, Ramat-Gan, Israel
[2] Racah Institute of Physics, The Hebrew University, 9190401 Jerusalem, Israel

The exact solution to the Boltzmann equation may be hard to achieve, so in many cases, an approximate solution is been used instead.
The most well-known approximation, the classic diffusion approximation can be developed by assuming that the angular dependence is isotropic or close to isotropic (alternatively, it is the most simple expression of the PN approximation, forcing N=1).
However, the diffusion approximation fails to describe the particles' density in highly anisotropic media.
A well-known modification for this approximation is the em asymptotic diffusion approximation.
This approximation is based on an asymptotic derivation in space and it successfully solves time-independent problems, yielding the correct asymptotic (infinite-media) transport eigenvalues. Nevertheless, in time-dependent problems, both the classic and asymptotic diffusion approximations fail to describe the particle's front density, due to the parabolic nature of the diffusion equation which predicts an infinite velocity for the particles.
Recently, a new approximation was proposed namely the asymptotic P1 approximation that is based on an asymptotic derivation in both space and time, which is a time-dependent equivalent of the asymptotic diffusion approximation. This approximation yields the correct time-independent eigenvalue of the exact transport equation, and the (almost) correct particle velocity.
However, there are still, non-negligible deviations between the asymptotic P1 approximation and the exact transport solutions, especially in anisotropic scenarios, due to the low order of this approximation. In our work, we (i) generalized the asymptotic P1 methodology for any given N, deriving the time-dependent asymptotic PN approximation, and (ii) show that it converges much faster than the classic PN approximation for any given N to the exact solution, in a 1D benchmark.