Positive Filtered PN Moment Closures for Linear Kinetic Equations

Abstract. We propose a positive-preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FPN ) expansion in the angular variable. The recently proposed FPN moment equations are known to suffer from the occurrence of (unphysical) negative particle concentrations. The origin of this problem is that the FPN 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 pre-selected 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 as the moment order tends to infinity, the FP N approximation converges, in the L sense, at the same rate as the FPN approximation; numerical tests suggest that this assumption may not be necessary. For purposes of comparison, we also consider a positive-preserving UDN closure that is based on the uniform damping of coefficients in the FPN approximation. While simple and less expensive to implement, the UDN approximation does not converge as fast as the FPN approximation for problems with limited regularity. We simulate the challenging line source benchmark problem with moment equations using several different choices of closure. The line source results indicate that, when compared to the UDN closure, the accuracy of the FP + N closure makes up for the overhead incurred by the optimization problem. In addition, we observe that for a regularized version of the line source problem, the UDN closure causes severe degradation in the space-time convergence of the PDE solver, while the FP N closure does not.