PaperlayerView exact revision on arXiv

Source-linked AI summary

Uniform convergence of diffusion synthetic acceleration for heterogeneous slab transport

Matthias Schlottbom

arXiv:2609.31346v1math.NAcs.NA

TL;DR

Source iteration is slow in optically thick, scattering-dominated media, and Fourier-based convergence results apply to infinite homogeneous media. This paper analyzes heterogeneous slabs and variational discretizations, proving a uniform convergence bound and a bounded preconditioned-system condition number.

  • Problem

    Source iteration slows when scattering dominates, while established Fourier convergence rates concern infinite homogeneous media rather than heterogeneous slabs with inflow boundaries.

    Source: §1 · p. 1

  • Method

    The authors use a min–max characterization of the DSA spectral radius and verify a sufficient rate condition with an energy-stable weighted angular average.

    Source: §4 · p. 8

  • Results

    The infinite-medium rate uniformly bounds the DSA spectral radius for heterogeneous slabs and their discretizations, and the preconditioned-system condition number is bounded by 1 + 0.29 c.

    Source: §1 · p. 3, §6 · p. 18

  • Takeaways & Limitations

    The bound is independent of slab thickness, spatial mesh, and angular resolution, and implies rapid convergence of preconditioned conjugate gradients.

    Source: §6 · p. 18

    In multiple dimensions, the local weighted projection does not control the full spatial gradient, so a bound of this form would require a spatially nonlocal interpolant.

    Source: §6 · p. 18

Abstract

from arXiv · show

The diffusion synthetic accelerated (DSA) source iteration is a standard solver for the radiative transfer equation. Using Fourier analysis, a convergence rate \(\rho_\infty(c)\le0.2247\,c\) with \(c\) the maximum ratio of scattering to total cross section has been established for an infinite homogeneous medium. For slab geometry with inflow boundary conditions and arbitrary bounded cross sections we prove that the spectral radius of the DSA iteration is at most \(\rho_\infty(c)\). We show that this convergence rate carries over to a variational discretization of the DSA iteration on every conforming tensor-product Galerkin space whose angular factor contains the constants. The analysis rests on an exact min--max characterization of the spectral radius. From it we derive a checkable sufficient condition for a given rate. We verify this condition using a suitable energy-stable projection, a weighted angular average whose weight is chosen so that the condition holds with the rate \(\rho_\infty(c)\). Since the projection maps discrete spaces into discrete spaces, the argument applies verbatim to the discrete iteration. As a consequence, the condition number of the preconditioned system is uniformly bounded by \(1+0.29\,c\), which implies rapid convergence of the preconditioned conjugate gradients method.

1. Introduction.

Source iteration slows in optically thick, scattering-dominated media, motivating diffusion synthetic acceleration (DSA). This paper establishes a uniform convergence bound for heterogeneous slabs and a class of variational discretizations.

Motivation

  • Source iteration converges slowly in optically thick media because the scattering-to-total-cross-section ratio c is close to one.Each sweep reduces scalar-flux error by at least the factor c in a suitable norm.Source: §1 · p. 1

Background

  • DSA corrects each sweep with a diffusion solve, targeting the slow convergence of source iteration in the diffusion regime.Fourier analysis for an infinite homogeneous medium gives ρ∞(c) ≤ 0.2247 c.Source: §1 · pp. 1–2

Convergence theory

  • Earlier finite-slab analyses either predict no convergence rate in limiting regimes or require optically thick cells, whereas the bounds here hold on every mesh.Prior variational work reported a rate c despite observed robustness in the diffusive regime.Source: §1 · p. 2

Approach and main results

  • The DSA spectral radius is bounded by the infinite-medium rate for heterogeneous slabs and their conforming variational discretizations, independently of slab thickness, mesh size, and angular resolution.The scalar-flux error is reduced by at least the factor ρ∞(c) per DSA step.Source: §1 · p. 3
  • The analysis uses an exact min–max characterization and verifies a sufficient rate condition with an energy-stable weighted angular average that also preserves discrete spaces.The paper's organization places the formulation and discretization before the spectral analysis and construction of the weighted average.Source: §1 · p. 3

2. Variational setting.

The variational setting uses an even-parity transport formulation to define the DSA sweep and diffusion correction. Its discretization is compatible with the analysis on arbitrary meshes when the angular space contains constants.

Even-parity formulation

  • The even-parity formulation yields the scalar flux by angular averaging, and the variational DSA iteration performs a transport sweep followed by a diffusion correction.The diffusion problem's Marshak boundary conditions arise naturally from the formulation.Source: §2 · p. 4

Variational setting

  • The analysis assumes bounded nonnegative scattering and total cross sections with absorption bounded below by γ > 0, making the variational forms bounded, symmetric, and coercive.The absorption is σa = σt − σs.Source: §2 · p. 4

Discretization

  • The discrete spaces use an arbitrary spatial mesh and a finite-dimensional even angular space containing constants; the weighted angular averages used in the proof map discrete spaces into discrete diffusion spaces.The discrete forms are restrictions of the continuous forms.Source: §2 · pp. 4–5

3. Analysis of the DSA iteration.

The analysis characterizes the DSA spectral radius through a min–max principle and reduces a rate bound to a verifiable condition. That condition must be established with a projection that respects the discrete spaces.

3. Analysis of the DSA iteration

  • The DSA error operator is self-adjoint in a natural weighted inner product, enabling an exact min–max characterization of its spectral radius.The sweep alone has rate c; the analysis quantifies the diffusion correction to obtain improved bounds.Source: §3 · p. 5

3.1. A primal criterion for the rate

  • A lower bound on the energy quotient gives an upper bound on the spectral radius, and the resulting rate condition is necessary for rates between zero and one.The condition can be applied to the discrete iteration when its construction preserves discrete spaces.Source: §3.1 · pp. 6–7
  • A sufficient rate bound follows by proving X(f) ≤ r m(f) + (1 − r)Y(f), which yields ρ(G) ≤ r.The same criterion applies to the discrete iteration when the projection maps discrete spaces into their diffusion subspaces.Source: §3.1 · p. 7
  • The criterion carries over to discrete iterations, but continuous spectral-radius bounds do not automatically transfer because the discrete energy functionals are not restrictions of the continuous ones.The quotient in the min–max characterization prevents a direct comparison between the continuous and discrete spectral radii.Source: §3.1 · p. 8
  • The angular average of a transport-space function need not lie in the diffusion space, so the unweighted average cannot serve as the required interpolant.Instead, the analysis introduces a weighted angular average as a projection into the diffusion subspace.Source: §4 · p. 8

4. The main result.

A weighted angular average verifies sufficient conditions for DSA spectral-radius bounds at the infinite-medium rate, with the argument carrying over to the discrete iteration. These bounds support conjugate gradients and quantify the accuracy needed for inexact correction.

4. The main result.

  • The weighted angular average Λwv = P(wv) is an analysis device; neither it nor its weight changes Algorithm 2.1 or its discretization.The passage states the weight assumptions w ≥ 0, w even, w not identically zero, and w/µ ∈ L2(−1, 1).Source: §4 · p. 8

4.1. Conditions for the weight.

  • Theorem 4.3 bounds the iteration rate by r when αC∂(w) ≤ 1, αCT(w) ≤ 1, and Qr(g, g) ≤ 0 for all g; the same conditions apply to the discrete iteration.A simple universal weight gives the robust bound c/4, while a c-dependent weight recovers ρ∞(c).Source: §4.1 · pp. 9–10
  • The weight w = 2β|µ|, with β = 3/(4 − c) and r = c/4, satisfies the sufficient conditions for the rate bound.The two eigenvalues used to verify the collision-form condition are c/4 and 3c/(4(4 − c)).Source: §4.1 · p. 10

4.2. A weight yielding ρ∞(c).

  • Choosing wλ with λ a global maximizer of ωc satisfies the weight conditions with r = ρ∞(c), including αCT(wλ) = 1 and αC∂(wλ), αCA(wλ) ≤ 1.Theorem 4.3 then gives ρ(G) ≤ r and ρ(Gh) ≤ r.Source: §4.1 · p. 10, §4.2 · p. 13
  • For λ ≥ λ1, h(λ) ≥ 0 ensures the Qr-condition; equivalently, the weighted average does not amplify constants, and at c = 1 it is a projection.The selected weight also makes CT(wλ) dominate both competing stability constants.Source: §4.2 · pp. 14–15
  • The weighted-average argument yields the spectral-radius bounds for both G and Gh, and its admissibility on discrete spaces requires only that they contain constants.The discrete iteration therefore inherits the same bounds.Source: §4.2 · p. 15
  • The preconditioned spectrum lies in [1 − ρ∞(c), 1], giving condition number at most 1.29 and enabling conjugate gradients for the transport system.The same bounds hold for the discrete iteration.Source: §4.2 · p. 15
  • The experiments use continuous P1 finite elements in space and piecewise-constant angular cells to examine the discrete bound, its sharpness for heterogeneous media, and conjugate-gradient benefits.Spectral radii and constants are computed as generalized eigenvalues.Source: §5 · p. 15

5. Numerical experiments.

The numerical experiments support the uniform bound across heterogeneous media and discretizations, show when computed rates approach it, and report reduced sweep counts with preconditioned conjugate gradients.

Experimental setup

  • The tests include piecewise-constant layered media, periodic media, and 120 random realizations spanning one to five layers and c values from 0.3 to 0.999.The listed piecewise-constant test media have c=0.9999.Source: §5 · pp. 15–16, Table 1 · p. 16

The bound ρ∞(c)

  • On graded meshes, ρ(Gh) approaches ρ∞(c)=0.22463 from below, mainly with angular refinement; uniform meshes need J≥800 to return 0.22458.The least-damped modes have wavelength about 2.5 mean free paths, which graded meshes resolve with cells of about one mean free path in the thick region.Source: Table 2 · p. 16, §5 · p. 16
  • Across 40 random heterogeneous media per scattering ratio, no computed rate violates the bound; for N=1, the diffusion correction is exact and ρ(Gh)=0.qh≤1 as asserted by the proof, while qh near one reflects saturation of the transport-term estimate, not attainment of the global rate.Source: §5 · pp. 16–17, Table 3 · p. 17

Sharpness for heterogeneous media

  • ρ(Gh) reaches ρ∞(c) up to discretization error when a thick layer attains c; periodic layers remain 0.22275–0.22292 versus ρ∞(0.999)=0.22434.Across the random ensemble, the largest ρ(Gh)/ρ∞(c) is 0.9995; these observations do not establish equality for an individual slab or optimality of the weight.Source: Table 4 · p. 17, §5 · p. 17

Discretization effects

  • For ε≤10^-3, optically thick uniform cells omit the slow modes and kinetic boundary layers, producing small discrete rates that are not a property of the continuous iteration.On meshes resolving those layers, the rate is close to ρ∞(c).Source: §5 · p. 17

Conjugate gradients

  • Preconditioned conjugate gradients needs at most seven iterations and reduces sweep counts by roughly one third to one half.Computed condition numbers stay below 1.29, reaching that bound in scattering-dominated cases.Source: §5 · p. 17, Table 5 · p. 18

6. Conclusions.

The infinite-medium rate ρ∞(c) uniformly bounds the DSA spectral radius for heterogeneous slabs and their stated discretizations, independent of slab thickness, mesh size, and angular resolution. The paper also identifies boundaries to extending the framework and points to related acceleration settings.

Conclusions

  • The infinite-medium value ρ∞(c) uniformly bounds the spectral radius for heterogeneous slabs and their discretizations, while the preconditioned condition number is bounded by 1+0.29 c.The estimates are independent of slab thickness, spatial mesh, and angular resolution.Source: §6 · p. 18
  • In multiple dimensions, the local weighted projection does not control the full spatial gradient, so a bound of this form would require a spatially nonlocal interpolant; anisotropic scattering also changes the projection structure.The authors note similar arguments may remain possible for weakly anisotropic scattering.Source: §6 · p. 18
  • The bounds may also be useful when source iteration is the inner iteration in diffusion-accelerated k-eigenvalue or Anderson- and Chebyshev-accelerated power methods.The cited applications are studied in and.Source: §6 · p. 18
Loading 2609.31346v1…