Convergence of Filtered Spherical Harmonic Equations for Radiation Transport

We analyze the global convergence properties of the filtered spherical harmonic (FPN ) equations for radiation transport. The well-known spherical harmonic (PN ) equations are a spectral method (in angle) for the radiation transport equation and are known to suffer from Gibbs phenomena around discontinuities. The filtered equations include additional terms to address this issue that are derived via a spectral filtering procedure. We show explicitly how the global L2 convergence rate (in space and angle) of the spectral method to the solution of the transport equation depends on the smoothness of the solution (in angle only) and on the order of the filter. The results are confirmed by numerical experiments. Numerical tests have been implemented in MATLAB and are available online.