# Angular adaptivity with spherical harmonics for Boltzmann transport

## TL;DR

This paper introduces an angular adaptivity algorithm for Boltzmann transport using Pn and filtered Pn expansions, which adaptively adjusts the expansion order and filter strength to improve efficiency and accuracy in complex transport problems.

## Contribution

The paper presents a novel angular adaptivity method combining Pn and filtered Pn expansions with spatially dependent filtering, reducing degrees of freedom and runtime while maintaining high-order convergence.

## Key findings

- Significant reduction in degrees of freedom and runtime.
- Adaptive filtered Pn achieves high-order convergence with localized heavy filtering.
- Method is competitive with P0 discretizations in advection-heavy problems.

## Abstract

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses Pn and filtered Pn expansions, allowing for different expansion orders across space/energy. Our spatial discretisation is specifically designed to use less memory than competing DG schemes and also gives us direct access to the amount of stabilisation applied at each node. For filtered Pn expansions, we then use our adaptive process in combination with this net amount of stabilisation to compute a spatially dependent filter strength that does not depend on a priori spatial information. This applies heavy filtering only where discontinuities are present, allowing the filtered Pn expansion to retain high-order convergence where possible. Regular and goal-based error metrics are shown and both the adapted Pn and adapted filtered Pn methods show significant reductions in DOFs and runtime. The adapted filtered Pn with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with P0 discretisations in problems with heavy advection.