On solutions to the Pn equations for thermal radiative transfer

We present results for the spherical harmonics (Pn) method for solving problems of time-dependent thermal radiative transport. We prove a theorem that demonstrates that in the streaming limit, the spatially and temporally continuous Pn equations will allow negative energy densities for any finite order of n. We also develop an implicit numerical method for solving the Pn equations to explore the impact of the theorem. The numerical method uses a high-resolution Riemann solver to produce an upwinded discretization. We employ a quasi-linear approach to integrate the nonlinearites added to make the scheme non-oscillatory. We use the backward Euler method for time integration and treat the material interaction terms fully nonlinearly. Reflecting boundary conditions for the Pn equations are presented and we show how to implement this boundary condition using ghost cells. The implicit method was able to produce robust results to thermal transport problems in one and two dimensions. The numerical method is used to analyze the accuracy of various Pn expansion orders on several problems. In two-dimensional problems the numerical Pn solutions contained negative radiation energy densities as predicted by our theorem. The numerical results showed that the material temperature also became negative, a result outside the scope of the theorem. Our numerical method can handle these negative values, but they would cause problems in a radiation-hydrodynamics calculation.