Diffusion approximations and domain decomposition method of linear transport equations: asymptotics and numerics

TL;DR

This paper develops numerical schemes to approximate linear transport equations with slab geometry using diffusion equations, handling both pure diffusive and mixed scalings through asymptotic analysis and boundary layer resolution.

Contribution

It introduces a unified approach for deriving diffusion approximations from linear transport equations with general scattering kernels and data, including boundary layer treatment.

Findings

Validated algorithms through numerical experiments.

Achieved error analysis for pure diffusive scaling.

Successfully handled mixed kinetic and diffusive scalings.

Abstract

In this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations. We treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist. The diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data. We apply the half-space solver in [20] to resolve the boundary layer equation and obtain the boundary data for the diffusion equation. The algorithms are validated by numerical experiments and also by error analysis for the pure diffusive scaling case.