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

TL;DR

This paper introduces a finite element method for angular discretization of the radiation transport equation on spherical geodesic grids, combining strengths of existing schemes and employing positivity-preserving strategies.

Contribution

A novel finite element approach on spherical geodesic grids that integrates advantages of $S_N$ and $FP_N$ methods, improving robustness and accuracy in radiation transport simulations.

Findings

Performs well in test problems where other methods fail

Combines strengths of $S_N$ and $FP_N$ schemes

Employs positivity-preserving limiting strategy

Abstract

Discrete ordinate ($S_N$) and filtered spherical harmonics ($FP_N$) 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 $FP_N$ schemes using four test problems and is found to perform well when one of the other methods fail.