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arXiv · 1807.06109

A Positive Asymptotic Preserving Scheme for Linear Kinetic Transport Equations

M. Paul Laiu·Martin Frank·Cory D. Hauck
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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.

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M. Paul Laiu, Martin Frank, Cory D. Hauck. 2018-07-16. A Positive Asymptotic Preserving Scheme for Linear Kinetic Transport Equations. https://arxiv.org/abs/1807.06109

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