On solutions to the Pn equations for thermal radiative transfer

A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations

“…The spherical harmonics method has several interesting characteristics. Due to hyperbolicity, the P N equations approximate radiation as a series of waves with velocity bounded by the speed of light [33]. This restriction is consistent with the transport equation, in contrast to diffusion methods where radiation propagates at infinite velocity.…”

“…Moreover, such a configuration favors negative solutions in the P N scheme. Indeed, the analytical P N solution to this problem was shown to have regions with negative values of the energy [9,33,32], while the P 1 solution even exhibits a negative singularity [30]. For all the results presented here, we use a grid with resolution ∆x = ∆y = 0.02 and a CFL factor of 0.0625.…”

“…These oscillations are related to the so-called Gibbs phenomenon that occurs when non-smooth functions are approximated with smooth basis functions [8]. The worst consequence of such oscillations is that they can cause the radiation intensity to become negative, which may lead to negative matter temperatures when the radiation transport is coupled to the matter energy equation [33,38]. There have been several attempts to address this problem [40,38,10,32,39,25].…”

Resolution of the time dependentPnequations by a Godunov type scheme having the diffusion limit

“…In 2D, the P n model, also known as the spherical harmonics method, preserves the rotational invariance of the neutron transport equation in contrast to the S n+1 model suffering from ray effects [18]. The P n model in 2D does not preserve the positivity of the density on the contrary of the S n+1 model: see [22] for a detailed study of the regimes when this problem may occur. Nevertheless, in some applications, the rotational invariance of the solution is more important than its positivity.…”

Second-order time evolution of PN equations for radiation transport

“…They derive a simplified P 3 that is strictly positive, but it is probably too diffusive and too specialized to be useful in practical calculations. More discussion of P N positivity is given in [9]. There are ways to reduce spatial oscillations by using a Riemann solver and upwind discretization or discontinuous Galerkin methods [6][7][8][9][10][11].…”