Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations
Abstract
Diffusion synthetic acceleration is most effective when its diffusion correction reflects the transport discretisation that generates the iteration error. We develop this principle for high-order upwind discontinuous Galerkin discretisations of discrete-ordinates transport on polytopic meshes. From the discrete transport sweep, we derive the exact scalar correction that removes the source-iteration scalar error in one step. We prove that the associated scalar response is positive and self-adjoint, obtain an exact expression for the source-iteration convergence factor, and quantify the additional damping produced by vacuum leakage. Using the exact correction as a reference, we construct a transport-matched modified interior penalty correction whose boundary terms are inherited directly from homogeneous vacuum inflow. In the optically thick regime, the resulting MIP form approximates the exact correction with relative error proportional to the effective cell Knudsen number. This gives a strict acceleration of source iteration, with bounds uniform in mesh size, polynomial degree, and element face count for admissible polytopic meshes. Numerical experiments on Cartesian and centroidal Voronoi meshes confirm the predicted convergence and correction-operator scaling.
Related
- Uniform convergence of diffusion synthetic acceleration for heterogeneous slab transport
- The discontinuous Galerkin method for solving the two-dimensional transport equation with distributed-order memory kernel on triangular meshes
- A super-convergent quasi-discontinuous Galerkin method for advection-diffusion problems
- A Fully Discrete Local Discontinuous Galerkin Method for Quasilinear Stochastic Convection-Diffusion-Type Equations
- A weighted polygonal discontinuous Galerkin method for hierarchically coupled reaction-diffusion systems with cubic interactions
BibTeX
@misc{calloo2026transportmatched,
title = {{Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations}},
author = {Ansar Calloo and Matthew Evans and François Madiot and Tristan Pryer},
howpublished = {HAL},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28022},
eprint = {2608.28022},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05738357v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations
AU - Ansar Calloo
AU - Matthew Evans
AU - François Madiot
AU - Tristan Pryer
PY - 2026
DA - 2026/08/28
PB - HAL
DO - 10.48550/arxiv.2608.28022
UR - https://hal.science/hal-05738357v1
AB - Diffusion synthetic acceleration is most effective when its diffusion correction reflects the transport discretisation that generates the iteration error. We develop this principle for high-order upwind discontinuous Galerkin discretisations of discrete-ordinates transport on polytopic meshes. From the discrete transport sweep, we derive the exact scalar correction that removes the source-iteration scalar error in one step. We prove that the associated scalar response is positive and self-adjoint, obtain an exact expression for the source-iteration convergence factor, and quantify the additional damping produced by vacuum leakage. Using the exact correction as a reference, we construct a transport-matched modified interior penalty correction whose boundary terms are inherited directly from homogeneous vacuum inflow. In the optically thick regime, the resulting MIP form approximates the exact correction with relative error proportional to the effective cell Knudsen number. This gives a strict acceleration of source iteration, with bounds uniform in mesh size, polynomial degree, and element face count for admissible polytopic meshes. Numerical experiments on Cartesian and centroidal Voronoi meshes confirm the predicted convergence and correction-operator scaling.
ER -