Fast Algorithm For Solving Time-dependent Multiscale radiative transport Equation

TL;DR

This paper introduces a fast, efficient algorithm for solving the time-dependent radiative transport equation by combining adaptive spatial discretization with a recursive skeleton method for operator inversion.

Contribution

The novel approach integrates adaptive finite point spatial discretization with a recursive skeleton method to efficiently solve multiple steady-state RTEs in time-dependent problems.

Findings

Achieves high accuracy in diverse scenarios.

Significantly reduces computational cost.

Effectively reconstructs layer structures after compression.

Abstract

When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying source terms, whose repeated solution is computationally costly. To address this, we first apply the adaptive tailored finite point scheme (TFPS) for spatial discretization. This scheme exploits prior knowledge of the background media's optical properties to adaptively compress the angular domain, constructing a compressed linear system. A key feature is its ability to reconstruct the layer structure after compression, faithfully capturing the variance at the layer. We then use the Recursive Skeleton Method (RSM) to obtain an explicit multilevel decomposition of the inverse discrete operator, which is reused for all steady-state solutions. Numerical experiments show that our framework achieves high accuracy and significant efficiency across diverse scenarios.