The second-order formulation of the $P_N$ equations with Marshak boundary conditions

TL;DR

This paper reformulates the classical $P_N$ method for linear transport equations into a second-order PDE system with Marshak boundary conditions, enabling flexible modeling and efficient numerical implementation.

Contribution

It introduces a second-order PDE reformulation of the $P_N$ equations with Marshak boundary conditions, supporting heterogeneous coefficients and complex geometries.

Findings

Effective handling of heterogeneous coefficients and irregular grids.

Demonstrated applicability through various numerical tests.

Implementation available online for rapid prototyping.

Abstract

We consider a reformulation of the classical $P_N$ method with Marshak boundary conditions for the approximation of the monoenergetic stationary linear transport equation as a system of second-order PDEs. Our derivation allows the automatic generation of a model hierarchy which can then be handed to standard PDE tools. This method allows for heterogeneous coefficients, irregular grids, anisotropic boundary sources and anisotropic scattering. The wide applicability is demonstrated in several numerical test cases. We make our implementation available online, which allows for fast prototyping.