Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions

Berninger, Heiko; Frénod, Emmanuel; Gander, Martin J.; Liebendörfer, M.; Michaud, Jérôme

doi:10.1137/12089243x

Cited by 8 publications

(25 citation statements)

References 53 publications

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“…This would be an alternative to the current version of the IDSA, which still has some mathematical issues that need to be fixed, see [2,3] for more details.…”

Section: Resultsmentioning

confidence: 99%

“…This assumption has been used at least in two different series of papers: the first one is in physics for the approximation of neutrino radiative transfer in core-collapse supernovae [11,2,3], and the second one is in mathematics for the coupling between the kinetic equation and approximations of it (diffusion, Euler, Navier-Stokes...) [8,5,6,7].…”

Section: Motivationmentioning

confidence: 99%

“…The methods of Degond et al [8,5,6,7] belong to the eFDDM class, but the IDSA [11,2,3] is an example of an iFDDM.…”

Section: Remarkmentioning

confidence: 99%

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Fuzzy Domain Decomposition: A New Perspective on Heterogeneous DD Methods

Gander

Michaud

2014

Lecture Notes in Computational Science and Engineering

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In a wide variety of physical problems, the complexity of the physics involved is such that it is necessary to develop approximations, because the complete physical model is simply too costly. Sometimes however the complete model is essential to capture all the physics, and often this is only in part of the domain of interest. One can then use heterogeneous domain decomposition techniques: if we know a priori where an approximation is valid, we can divide the computational domain into subdomains in which a particular approximation is valid and the topic of heterogeneous domain decomposition methods is to find the corresponding coupling conditions to insure that the overall coupled solution is a good approximation of the solution of the complete physical model. For an overview of such techniques, see [9,10] and references therein. However, there are many physical problems where it is not a priori known where which approximation is valid. In such problems, one needs to track the domain of validity of a particular approximation, and this is usually not an easy task. An example of such a method is the -method, see [4,1].In this contribution, we introduce a new formalism for heterogeneous domain decomposition, which is not based on a sharp decomposition into subdomains where different models are valid. The main idea relies on the notion of Fuzzy Sets introduced by Zadeh [12] in 1965. The Fuzzy Set Theory relaxes the notion of belonging to a set through membership functions to (fuzzy) sets that account for partially belonging to a set. In the context of heterogeneous domain decomposition, this could be useful if one assumes that the computational domain can be decomposed into fuzzy sets that form a partition of the domain in a sense that needs to be specified. Once such a partition is given, one can compute the solution of the coupled problem using the membership functions. Note that the membership functions can depend on space and time and therefore can take into account a change in the validity domain of a particular approximation. We show here that this technique leads to an excellent coupling strategy for the 1D advection dominated diffusion problem. Such a domain decomposition method would be able, in principle, to take into account part of the domain where none of the available approximations are valid under the assumption that a combination of them is a good enough approximation there.On the assumption u = u 1

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“…This would be an alternative to the current version of the IDSA, which still has some mathematical issues that need to be fixed, see [2,3] for more details.…”

Section: Resultsmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

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Fuzzy Domain Decomposition: A New Perspective on Heterogeneous DD Methods

Gander

Michaud

2014

Lecture Notes in Computational Science and Engineering

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“…The reaction limit is valid in regions where the mean free path κ −1 is negligible compared with the typical length scale of the system. As for the diffusion limit, one can use asymptotic expansion with a different scaling in order to derive this limit, see for example [2]. 6 JÉRÔME MICHAUD with g idsa := h free .…”

mentioning

confidence: 99%

“…In [2], it is shown that this particular coupling mechanism might be problematic and mathematical issues are discussed. The aim of this paper is to study the behavior of the IDSA on a simple radiative transfer model and try to address some of the mathematical issues highlighted in [2].…”

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confidence: 99%

The IDSA and the homogeneous sphere: Issues and possible improvements

Michaud¹

2016

DCDS-S

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In this paper, we are concerned with the study of the Isotropic Diffusion Source Approximation (IDSA) [6] of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA which is numerically shown to be more accurate than the old one. Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation. 4 JÉRÔME MICHAUD 2.2. Reaction limit. The reaction regime is a special limit of the diffusion limit in which the total opacity κ is so large that the diffusion term can be dropped in Eq. (9). This corresponds to the following closure relationsThe corresponding evolution equation readswhere we defined the reaction evolution operator L reac J for J as before. The reaction limit is valid in regions where the mean free path κ −1 is negligible compared with the typical length scale of the system. As for the diffusion limit, one can use asymptotic expansion with a different scaling in order to derive this limit, see for example [2]. 6 JÉRÔME MICHAUD with g idsa := h free . Comparing this closure relations with the asymptotic limits described in Section 2, we have the following operator definitions L t J := L reac J Received xxxx 20xx; revised xxxx 20xx.

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Partition of Unity Methods for Heterogeneous Domain Decomposition

Ciaramella

Gander

2018

Lecture Notes in Computational Science and Engineering

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Heterogeneous problems and partition of unity decompositionWe are interested in solving linear PDEs of the formwhere Ω is a bounded domain in R d with d = 1, 2, L is a linear (elliptic) differential operator, f and g are the data, and u is the solution to (1). The weak form of (1) with a Hilbert spacewhereis the bilinear form corresponding to the operator L , and : V → R is the linear functional induced by f . We assume that (2) has a unique solution u ∈ {v ∈ V : v = g on ∂ Ω }, and that u is "heterogeneous", behaving very differently in different parts of Ω . Typical examples are advection-diffusion problems, where there are advection dominated and diffusion dominated regions (subdomains), and the boundaries in between are not clearly defined, see [8,10] and references therein. Apart from the χ-method [6, 1], there are no methods to determine such subdomain decompositions, and our goal is to present and study a new such method. We thus introduce (see [18,11])Given two membership functions ϕ 1 , ϕ 2 : Ω → [0, 1] that form a partition of unity on Ω , ϕ 1 (x) + ϕ 2 (x) = 1 for all x ∈ Ω , their supports provide then a domain decompo-

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Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions

Cited by 8 publications

References 53 publications

Fuzzy Domain Decomposition: A New Perspective on Heterogeneous DD Methods

Fuzzy Domain Decomposition: A New Perspective on Heterogeneous DD Methods

The IDSA and the homogeneous sphere: Issues and possible improvements

Partition of Unity Methods for Heterogeneous Domain Decomposition

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