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Transport-Matched Penalties for Diffusion Synthetic Acceleration of Polytopic Discontinuous Galerkin Discretisations
Ansar Calloo, Matthew Evans, Francois Madiot, Tristan Pryer
TL;DR
Optically thick scattering makes source iteration ineffective, motivating diffusion corrections compatible with the transport discretisation. The paper derives an exact scalar correction from the upwind DG transport sweep and constructs a vacuum-matched MIP approximation. In the optically thick regime, the MIP correction yields strict acceleration with bounds uniform in mesh size, polynomial degree, and face count under admissible mesh assumptions.
Problem
Source iteration converges slowly in highly scattering, optically thick regimes, and effective DSA requires a correction compatible with the discrete transport operator.
Method
The paper derives the exact scalar correction induced by high-order upwind DG transport sweeps and constructs a transport-matched MIP correction whose boundary terms follow from homogeneous vacuum inflow.
Results
The MIP correction approximates the exact correction with relative error controlled by the effective cell Knudsen number and gives uniform accelerated-contraction bounds under admissible polytopic-mesh estimates.
Takeaways & Limitations
Vacuum-matched transport corrections provide strict source-iteration acceleration while remaining uniform in mesh size, polynomial degree, and element face count within the stated mesh class.
Takeaways & Limitations
The numerical experiments use bounded-domain vacuum conditions, while standard weak Dirichlet and Marshak diffusion boundaries are deferred to a companion study.
Abstract
from arXiv · showhide
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.
1. Introduction
Source iteration becomes ineffective in optically thick, highly scattering transport because slowly decaying scalar-error modes drive its convergence factor toward unity. The paper develops and analyses transport-compatible DSA corrections for high-order upwind DG discretisations on admissible polytopic meshes.
- Optically thick scattering makes source iteration increasingly ineffective as dominant scalar-error modes decay slowly and the convergence factor approaches unity.
- DSA accelerates transport iteration by correcting the scalar error with a diffusion operator compatible with the discrete transport operator.The paper emphasizes that diffusion stability alone does not guarantee effective acceleration.
- Polytopic meshes can have many strongly varying faces, motivating whole-boundary estimates whose constants are uniform in face count and relative face sizes.
- The paper identifies the exact scalar correction generated by the discrete transport sweep and analyses its approximation by a local MIP correction on admissible polytopic meshes.This incorporates polynomial degree, mesh geometry, and physical boundary conditions into the analysis.
- The MIP correction reproduces vacuum-inflow boundary terms and approximates the exact correction with uniform bounds in mesh size, polynomial degree, and element face count.The relative approximation is established in the optically thick regime.
- The study focuses on analytical operator quantities, including the exact source-iteration factor, accelerated contraction estimate, and relative discrepancy between corrections.A companion study examines broader computational settings and costs.
2. Transport problem and diffusion correction
The paper formulates steady monoenergetic transport with isotropic scattering under homogeneous vacuum inflow and derives a diffusion correction from the source-iteration error equations. Its boundary contribution is taken directly from the upwind DG zero-exterior-trace treatment rather than from a separate diffusion boundary closure.
- The model uses steady monoenergetic transport with isotropic scattering, constant coefficients, and homogeneous vacuum inflow on a bounded polytopal domain.
- Source iteration advances a scalar iterate through a directional transport predictor, whose error satisfies a corresponding transport equation.
- In the optically thick regime, slowly varying error is approximately isotropic, so angular moments motivate an interior diffusion correction.
- Vacuum transport does not provide a closed local diffusion boundary condition from moment equations alone without an additional boundary-layer approximation.
- The discrete correction instead obtains its boundary terms directly from the zero exterior trace in the upwind DG transport form.This produces a vacuum-matched MIP boundary contribution for algebraic comparison with the exact discrete correction.
3. DG discretisation on polytopic meshes
The spatial method uses upwind DG on admissible polytopic meshes with whole-boundary estimates designed to remain uniform under complex face structures. Its MIP diffusion form combines elliptic coercivity with angularly averaged transport dissipation and vacuum-matched boundary terms.
- Mesh assumptions: The mesh family consists of finite, connected, face-to-face partitions into bounded Lipschitz polytopes, with uniform geometric assumptions but no required convexity.
- Mesh assumptions: The analysis uses degree-explicit inverse and whole-boundary trace estimates with constants independent of mesh size, polynomial degree, and face count.These estimates are the mechanism for uniform operator and contraction bounds.
- Mesh assumptions: Admissibility is an element-level geometric property permitting many faces, small faces, suitable agglomerations, and non-convex elements when covering and star-shapedness constants remain uniform.
- Mesh assumptions: Arbitrary polytopic agglomerations and families with unbounded aspect ratio are excluded because degeneration can destroy the uniform trace and inverse estimates required by the analysis.
- Mesh assumptions: A uniform Voronoi separation-to-covering-radius bound provides admissibility for the corresponding Voronoi mesh family.
- Operator properties: The upwind form is symmetric in its angularly averaged representation and includes jump dissipation, while the MIP form is symmetric positive definite.
- MIP diffusion form: The MIP penalty combines the usual SIP penalty, which guarantees coercivity, with angularly averaged upwind jump dissipation for transport matching.
- MIP diffusion form: Vacuum-matched MIP boundary terms use half-factors inherited from the zero exterior average and are required for second-order matching with the exact vacuum correction.The form is distinct from full-flux Nitsche and Marshak boundary conditions.
4. Source iteration and DSA
The exact scalar correction induced by the transport sweep removes source-iteration scalar error in one step, while vacuum-matched MIP-DSA approximates it and strictly accelerates iteration in the optically thick regime.
- Exact scalar correction: The scalar response operator is self-adjoint and positive definite, making it invertible and defining the exact transport-induced scalar correction.It maps a common normalised scalar source to the scalar flux from the directional transport solves.
- Exact scalar correction: Solving the exact correction equation with the source-iteration defect returns the predictor error and removes the scalar error in one step.The exact correction form is symmetric positive definite and supplies the comparison energy for MIP-DSA.
- MIP replacement: MIP-DSA replaces the exact scalar correction form with a local diffusion form, so the corrected error is governed by their form difference.The MIP boundary treatment reproduces the vacuum-matched terms inherited from homogeneous vacuum inflow.
- Contraction estimates: The relative exact–MIP correction error is proportional to Knϵ,h,p, with constants independent of mesh size, polynomial degree, and element face count.The estimate applies under the optically thick condition where the transport floor is active and yields uniform equivalence of the exact and MIP energies.
- Contraction estimates: The exact source-iteration propagator is self-adjoint in the exact-correction inner product and has a contraction factor strictly below the coefficient-only bound cϵ.The same energy enables direct comparison between source iteration and MIP-DSA contraction factors.
- Contraction estimates: Within the optically thick regime, MIP-DSA is a strict acceleration of source iteration, whose diffusive-limit convergence factor approaches unity.The admissible-mesh framework permits polytopes with many faces when element-level geometric constants remain uniform.
5. Proofs of the discrete results
The proofs establish well-posed directional transport, a self-adjoint positive scalar response, and exact source-iteration contraction properties. Mesh- and degree-uniform estimates then control the MIP approximation through the effective cell Knudsen number.
- Directional transport: The directional transport problems are uniquely solvable under the stated positivity and vacuum-boundary conditions.The proof uses positivity of the directional energy identity and the zero exterior trace on physical boundaries.
- Scalar response: The normalised scalar response Hϵh is self-adjoint, positive definite, and bounded by the identity in the transport inner product.Central pairing supplies self-adjointness, while the quadratic estimate gives 0 ≺ Hϵh ⪯ I.
- Source iteration: Vacuum leakage makes the source-iteration contraction inequality strict, yielding a factor strictly below the coefficient-only bound cϵ.The exact correction identity is used to derive the strict contraction result under homogeneous vacuum inflow.
- Uniform estimates: The uniform streaming estimate has constant CB independent of h, p, and the number of element faces.This is the mesh-structure-independent ingredient in the macro–micro estimate.
- MIP approximation: The exact and MIP correction forms differ by a relative amount controlled by the effective cell Knudsen number, with constants independent of h, ϵ, p, and face count.The proof combines the exact second-order matching identity with lifting and remainder estimates.
6. Numerical verification under vacuum inflow
The vacuum-inflow experiments verify the exact source-iteration prediction and strict MIP–DSA improvement. They also confirm linear correction-form scaling with effective cell Knudsen number across mesh families and polynomial degrees.
- Experimental setup: The experiments evaluate exact and accelerated contraction factors together with the relative discrepancy between exact and vacuum-matched MIP corrections.They use homogeneous vacuum inflow and compare the quantities appearing in the analytical estimates.
- Experiment 1: Figure 1 confirms the exact source-iteration identity to numerical precision and shows strict improvement from MIP–DSA.The accelerated factors decrease with effective Knudsen number, while the accelerated spectral radius remains below the exact operator norm.
- Experiment 2: The correction-form discrepancy scales linearly with the effective Knudsen number, as predicted by Theorem 4.6.The scaled quantity remains bounded across polynomial degrees and mesh families.
- Experiment 2: Cartesian and centroidal Voronoi meshes show comparable scaled-discrepancy behaviour across polynomial degrees p = 1, 2, 3, 4.The second experiment tests both mesh families and the four listed polynomial degrees.
7. Conclusion
The paper constructs a transport-matched MIP correction from the exact upwind-DG scalar correction, including vacuum-induced boundary terms. Its relative error and accelerated-iteration bounds are controlled by the effective cell Knudsen number and uniform in key discretisation parameters.
- Conclusion: The exact correction and MIP form share vacuum-boundary structure through averaged leakage and half-weighted symmetric flux terms.Both arise from the zero exterior trace imposed on physical boundary faces.
- Conclusion: The relative form estimate yields a complete accelerated-iteration contraction bound uniform in mesh size, polynomial degree, and face count.The estimate is controlled by Knϵh,p under the stated whole-boundary assumptions.
- Conclusion: Under vacuum inflow, the exact source-iteration factor is cϵλmax(Hϵh), with strict reduction attributed to boundary leakage.The MIP–DSA estimate remains controlled by the effective cell Knudsen number.
- Scope: The numerical verification is restricted to bounded-domain vacuum formulations; weak Dirichlet and Marshak conditions are treated in companion computational work.The reported experiments verify the predicted interior operator scaling for the analysed vacuum-matched form.