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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 5

### Convergence of filtered spherical harmonic equations for radiation transport

Pages: 1443 – 1465

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n5.a10

#### Authors

#### Abstract

We analyze the global convergence properties of the filtered spherical harmonic $(\mathrm{FP}_N)$ equations for radiation transport. The well-known spherical harmonic $(\mathrm{P}_N)$ 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 $L^2$ 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.

#### Keywords

spherical harmonics, radiation transport

#### 2010 Mathematics Subject Classification

33C55, 65M70, 82D75