# Superconvergence Extraction of Upwind Discontinuous Galerkin Method Solving the Radiative Transfer Equation

## TL;DR

This paper analyzes the superconvergence properties of the upwind discontinuous Galerkin method for radiative transfer equations and demonstrates how SIAC filters significantly improve accuracy and computational efficiency.

## Contribution

It provides a theoretical proof of superconvergence orders and shows how SIAC filters enhance accuracy and efficiency in solving radiative transfer equations.

## Key findings

- Superconvergence orders of (2k+2) and (2k+1/2) proven for steady-state and time-dependent problems.
- SIAC filters improve accuracy by approximately 2.22 times for time-dependent and 4-9 times for steady-state problems.
- Filtering reduces computational time by nearly 20 times for time-dependent problems without accuracy loss.

## Abstract

We theoretically analyze the superconvergence of the upwind discontinuous Galerkin (DG) method for both the steady-state and time-dependent radiative transfer equation (RTE), and apply the Smooth-Increasing Accuracy-Conserving (SIAC) filters to enhance the accuracy order. Direct application of SIAC filters on low-dimensional macroscopic moments, often the quantities of practical interest, can effectively improve the approximation accuracy with marginal computational overhead. Using piecewise $k$-th order polynomials for the approximation and assuming constant cross sections, we prove $(2k+2)$-th order superconvergence for the steady-state problem at Radau points on each element and $(2k+1/2)$-th order superconvergence for the global $L^2$ and negative-order Sobolev norms for the time-dependent problem. Numerical experiments confirm the efficacy of the filtering, demonstrating post-filter convergence orders of $2k+2$ for steady-state and $2k+1$ for time-dependent problems. More significantly, the SIAC filter delivers substantial gains in computational efficiency. For a time-dependent problem, we observed an approximately $2.22 \times$ accuracy improvement and a $19.94 \times$ reduction in computational time. For the steady-state problems, the filter achieved a $4$--$9 \times$ acceleration without any loss of accuracy.