The preprint examines a modified diffusion correction for high-order discontinuous Galerkin transport and reports a strict acceleration of source iteration in the covered optically thick regime. The theorem ties the accelerated error to the effective cell Knudsen number, while numerical experiments found normalized accelerated factors decreasing with that number and the exact-to-MIP correction discrepancy following the predicted linear scaling.
The work concerns source iteration for steady monoenergetic transport with isotropic scattering and constant coefficients. Its diffusion synthetic acceleration, or DSA, supplies a scalar correction intended to approximate the exact scalar correction generated by the transport sweep. Within the discrete model, the exact correction removes the source-iteration scalar error in one step; the MIP version is the approximation under study.
The boundary changes the convergence picture
Before comparing the two corrections, the analysis characterizes a normalized scalar response for the discrete transport sweep. The response is self-adjoint and positive definite, and is bounded above by the identity operator. With homogeneous vacuum inflow and a nonempty physical boundary, it is strictly below the identity, so its largest eigenvalue is less than 1. This gives an additional reduction relative to a coefficient-only bound under vacuum inflow.
The exact source-iteration factor is the scattering ratio multiplied by the largest response eigenvalue. Its norm in the exact-correction energy equals its spectral radius, which measures the iteration's error amplification, and is no greater than the scattering ratio. Under homogeneous vacuum inflow, that inequality is strict.
The MIP correction is assembled face by face. On each mesh face, its penalty is the larger of the transport scaling parameter times the SIP penalty and the angularly averaged upwind jump-dissipation coefficient. The paper assigns the first term to coercivity and the second to matching the transport operator.
At a physical boundary, the construction uses a transport-matched vacuum form. Its one-half factors are inherited from the zero exterior average. The paper distinguishes this contribution from full-flux Nitsche and Marshak boundary conditions, so the boundary result is specific to the vacuum setting.
The central estimate applies when the effective cell Knudsen number is no greater than the theorem threshold. In that regime, the transport-floor penalty is active on every interior and physical-boundary face, and the MIP form is symmetric positive definite. The exact and MIP correction forms then differ by an amount controlled in the MIP energy norm by a constant times the effective cell Knudsen number. The bound is uniform over mesh size, transport scaling, polynomial degree and face count.
Numerical checks follow the theory
Numerical verification used a fixed positive, centrally paired angular quadrature. Unless otherwise stated, it contained 16 uniformly distributed directions with reciprocal weights.
In the first experiment, the exact source-iteration identity held to numerical precision. MIP-DSA showed a strict improvement over source iteration, and its normalized accelerated factors decreased with effective Knudsen number. The accelerated spectral radius remained below the corresponding exact operator norm. The reported account gives these as figure-based qualitative results rather than tabulated point estimates.
The second experiment computed the relative correction-form discrepancy through a generalized eigenvalue problem. It showed the predicted linear scaling with effective Knudsen number. After scaling by that number, the discrepancy remained bounded across the tested polynomial degrees and across Cartesian and centroidal Voronoi mesh families. The linear regime coincided with activation of the transport floor, including physical-boundary contributions.
A result with defined boundaries
The formal setting is steady monoenergetic transport with isotropic scattering, constant coefficients, homogeneous vacuum inflow, positive centrally paired quadratures and upwind DG on admissible polytopal meshes. The convergence analysis also assumes exact applications of the directional transport inverses and the MIP inverse, with directed dependencies, including cycles, resolved before a transport sweep.
Those conditions set the boundary of the conclusion: the theorem and tests address the stated vacuum, optically thick, fully discrete model, with exact operator applications assumed in the convergence analysis.
The document is an arXiv version 1 preprint dated 28 August 2026. The supplied front matter includes no funding statement.
Paper data and sources
Original title: Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations
Authors: Ansar Calloo, Matthew Evans, Francois Madiot, Tristan Pryer
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text