Positive Filtered P$_N$ Moment Closures for Linear Kinetic Equations

We propose a positive-preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP$_N$) expansion in the angular variable. The recently proposed FP$_N$ moment equations are known to suffer from the occurrence of (unphysical) negative particle concentrations. The origin of this problem is that the FP$_N$ approximation is not always positive at the kinetic level; the new FP$^+_N$ closure is developed to address this issue. A new spherical harmonic expansion is computed via the solution of an optimization problem, with constraints that enforce positivity, but only on a finite set of preselected points. Combined with an appropriate PDE solver for the moment equations, this ensures positivity of the particle concentration at each step in the time integration. Under an additional, mild regularity assumption, we prove that FP$^+_N$ has the same consistency as FP$_N$; that is, the FP$^+_N$ approximation converges to a given target function in $L^2$ at the same rate a...