Diffusion asymptotics for linear transport with low regularity

Egger, Herbert; Schlottbom, Matthias

doi:10.3233/asy-141235

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…For δ → 0, the corresponding solution u δ will converge to the solution of a diffusion problem; for nonsmooth coefficients see [20]. The parameter c defined in Lemma 3.5 is bounded by O(1/δ).…”

Section: Multi-d: the Lattice Problemmentioning

confidence: 99%

“…with parameters σ a = 1/100, σ s (z) = 2 + sin(πz)/2 and Z = 1, and source terms defined accordingly. We computed the numerical solution u h using the DSA preconditioned iteration (18), (19), (20). We stopped the iteration using the a-posteriori stopping rule…”

Section: Manufactured Solutionsmentioning

confidence: 99%

“…where both parametersσ s andσ a are bounded and strictly bounded away from zero. Since this is not the case for the lattice problem, we consider the following parameters For δ → 0, the corresponding solution u δ will converge to the solution of a diffusion problem; for nonsmooth coefficients see [20]. The parameter c defined in Lemma 3.5 is bounded by O(1/δ).…”

Section: Multi-d: the Lattice Problemmentioning

confidence: 99%

“…Among the most used and simple preconditioners is the diffusion synthetic acceleration method (DSA), in which a diffusion problem is solved in every iteration. This is well motivated by asymptotic analysis [19,20]. While Fourier analysis can be applied to special situations [16,17], the convergence analysis is mainly open for the general case, i.e., for jumping coefficients or non-periodic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

Palii

Schlottbom

2020

Computers & Mathematics with Applications

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We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinitedimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim.

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Section: Multi-d: the Lattice Problemmentioning

confidence: 99%

Section: Manufactured Solutionsmentioning

confidence: 99%

Section: Multi-d: the Lattice Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

Palii

Schlottbom

2020

Computers & Mathematics with Applications

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A Spherical Harmonic Discontinuous Galerkin Method for Radiative Transfer Equations with Vacuum Boundary Conditions

Sheng

Wang

2021

J Sci Comput

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Including scattering within the room acoustics diffusion model: An analytical approach

Foy

Picaut

Valeau

2016

The Journal of the Acoustical Society of America

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Over the last 20 years, a statistical acoustic model has been developed to predict the reverberant sound field in buildings. This model is based on the assumption that the propagation of the reverberant sound field follows a transport process and, as an approximation, a diffusion process that can be easily solved numerically. This model, initially designed and validated for rooms with purely diffuse reflections, is extended in the present study to mixed reflections, with a proportion of specular and diffuse reflections defined by a scattering coefficient. The proposed mathematical developments lead to an analytical expression of the diffusion constant that is a function of the scattering coefficient, but also on the absorption coefficient of the walls. The results obtained with this extended diffusion model are then compared with the classical diffusion model, as well as with a sound particles tracing approach considering mixed wall reflections. The comparison shows a good agreement for long rooms with uniform low absorption (α = 0.01) and uniform scattering. For a larger absorption (α = 0.1), the agreement is moderate, due to the fact that the proposed expression of the diffusion coefficient does not vary spatially. In addition, the proposed model is for now limited to uniform diffusion and should be extended in the future to more general cases.

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Diffusion asymptotics for linear transport with low regularity

Cited by 3 publications

References 23 publications

On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

A Spherical Harmonic Discontinuous Galerkin Method for Radiative Transfer Equations with Vacuum Boundary Conditions

Including scattering within the room acoustics diffusion model: An analytical approach

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