A Positive Asymptotic Preserving Scheme for Linear Kinetic Transport Equations

TL;DR

This paper introduces a positive, asymptotic preserving numerical scheme for linear kinetic transport equations that maintains positivity and accurately transitions to the diffusion limit, validated through benchmark tests.

Contribution

The paper develops a novel scheme combining spectral angular discretization, micro-macro decomposition, and realizability limiters to ensure positivity and asymptotic correctness.

Findings

Scheme preserves positivity of particle concentration.

Achieves correct diffusion limit as scattering becomes large.

Demonstrates expected accuracy on benchmark problems.

Abstract

We present a positive and asymptotic preserving numerical scheme for solving linear kinetic, transport equations that relax to a diffusive equation in the limit of infinite scattering. The proposed scheme is developed using a standard spectral angular discretization and a classical micro-macro decomposition. The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and realizability limiters in the angular discretization. Under mild assumptions on the initial condition and time step, the scheme becomes a consistent numerical discretization for the limiting diffusion equation when the scattering cross-section tends to infinity. The scheme also preserves positivity of the particle concentration on the space-time mesh and therefore fixes a common defect of spectral angular discretizations. The scheme is tested on the well-known line source benchmark problem with the usual uniform material medium as well as a medium composed from different materials that are arranged in a checkerboard pattern. We also report the observed order of space-time accuracy of the proposed scheme.