Communications in Mathematical Sciences

Volume 14 (2016)

Number 5

Convergence of filtered spherical harmonic equations for radiation transport

Pages: 1443 – 1465

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

Authors

Martin Frank (Department of Mathematics and Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany)

Cory Hauck (Applied and Computational Mathematics Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.; and Department of Mathematics, University of Tennessee, Knoxville, Tn., U.S.A.)

Kerstin Küpper (Department of Mathematics and Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany )

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

Full Text (PDF format)

Published 18 May 2016