A Finite Element Method for Angular Discretization of the Radiation Transport Equation on Spherical Geodesic Grids

Abstract

Discrete ordinate ($S_N$) and filtered spherical harmonics ($F{P}_N$) based schemes have been proven to be robust and accurate in solving the Boltzmann transport equation but they have their own strengths and weaknesses in different physical scenarios. We present a new method based on a finite element approach in angle that combines the strengths of both methods and mitigates their disadvantages. The angular variables are specified on a spherical geodesic grid with functions on the sphere being represented using a finite element basis. A positivity-preserving limiting strategy is employed to prevent non-physical values from appearing in the solutions. The resulting method is then compared with both $S_N$ and $F{P}_N$ schemes using four test problems and is found to perform well when one of the other methods fail.

24 pages, 13 figures

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