A discretization of the multigroup P radiative transfer equation on general meshes

Hermeline, F.

doi:10.1016/j.jcp.2016.02.058

Cited by 9 publications

(14 citation statements)

References 30 publications

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“…M = 1 and η = η 1 see (2)), homogeneous and considered uncertain. It depends on three parameters X = (X 1 , X 2 , X 3 ) affecting the total and scattering cross-sections and the material density as 31 Best compromise in term of relative accuracy and restitution times. To find it, we simply ran many tests for different…”

Section: Sensitivity Analysis In 3d Stochastic Dimensionmentioning

confidence: 99%

“…The relevance of the latter integrodifferential PDE is not to demonstrate. Amongst the applications (non exhaustive list), one can quote biology [58] with population dynamics, plasma physics (transport of ions and electrons) [21], photonics [64,50,17,41,31,52] or neutronics [68,40,22,33,34,23]. (ii) Second, gPC can be applied intrusively (see for examples [43,76,20]) or non-intrusively (the most common way) and whether one strategy performs better than the other is far from being obvious.…”

Section: Introductionmentioning

confidence: 99%

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A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

Poëtte

2019

Journal of Computational Physics

144

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In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a media whose composition is uncertain. The most common resolution strategy consists in running, at prescribed points in the uncertain space (at experimental designs points), a simulation device as a black-box and perform the uncertainty propagation resolution (i.e. resolution of SPDE). This kind of strategy is commonly called non-intrusive. The nonintrusive resolution can be carried with Monte-Carlo, quasi Monte-Carlo, generalized Polynomial Chaos etc. The latter is of interest in this document for its fast convergence rate. After going over and illustrating the main drawbacks of the non-intrusive (gPC or not) uncertainty propagation resolution for the linear Boltzmann equation in a simplified configuration, we build a new gPC based MC scheme solving intrusively the uncertain counterpart of the problem. The paper ends with some numerical examples.

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Section: Sensitivity Analysis In 3d Stochastic Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

Poëtte

2019

Journal of Computational Physics

144

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“…We adopt the common strategy which is to use write the P N and S N reduced models [8,17] in the form of Friedrichs system with relaxation, as in to (2). Numerical approximation of the transport equation and related reduced models is challenging because of the two spatially dependent coefficients σ a and σ s .…”

Section: Introductionmentioning

confidence: 99%

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P ₁ Model

Morel

Buet

Després

2018

Computational Methods in Applied Mathematics

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This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a {P_{N}} or {S_{N}} model, and we study in more details the {P_{1}} model in dimension 1 and 2. We show that the TDG method provides natural well-balanced and asymptotic preserving discretization since exact solutions are used locally in the basis functions. High-order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for {P_{1}} model in dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.

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“…This model can be motivated as an angular discretization of a linear Boltzmann equation which is a transport/kinetic equation where the physics is modeled with linear interaction relaxation operators (4). We adopt a convention, which comes from the physics of transfer or neutron propagation [22], that the matrices A 1 and A 2 have a block structure…”

Section: Introductionmentioning

confidence: 99%

“…A natural structure which models relaxation mechanisms is R + R t ≥ 0. In our work, we follow closely the convention proposed in [22] by taking…”

Section: Introductionmentioning

confidence: 99%

Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

2020

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The Trefftz discontinuous Galerkin (TDG) method provides natural well-balanced (WB) and asymptotic preserving (AP) discretization since exact solutions are used locally in the basis functions. However, one difficult point may be the construction of such solutions which is a necessary first step in order to apply the TDG scheme. This works deals with the construction of solutions to Friedrichs systems with relaxation with application to the spherical harmonics approximation of the transport equation (the so-called PN models). Various exponential and polynomial solutions are constructed. Two numerical tests on the P3 model illustrate the good accuracy of the TDG method. They show that the exponential solutions lead to accurate schemes to capture boundary layers on a coarse mesh and that a combination of exponential and polynomial solutions is efficient in a regime with vanishing absorption coefficient.

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A discretization of the multigroup P radiative transfer equation on general meshes

Cited by 9 publications

References 30 publications

A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P ₁ Model

Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

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A discretization of the multigroup P radiative transfer equation on general meshes

Cited by 9 publications

References 30 publications

A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P 1 Model

Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

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Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P ₁ Model