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Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations

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Clément Cancès (RAPSODI), Claire Chainais-Hillairet (RAPSODI), Stella Krell (COFFEE, UCA, JAD)

Analysis of PDEs · Submitted 30 May 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

how it is stated · asserted 0.99 read the record →

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arXiv ID1705.10558 (version 1)
Pith IDASMT4YWX, Pith's permanent record of this paper and its review
CitationClaire Chainais-Hillairet (RAPSODI), Cl\'ement Canc\`es (RAPSODI), JAD), Stella Krell (COFFEE, UCA. "Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations." arXiv:1705.10558 (2017). https://pith.science/paper/ASMT4YWX
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,419 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. contribution 0.96 0.94 0.11 asserted 0.99
s1 The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. contribution 0.56 0.87 0.16 asserted 1.00
s2 This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. interpretation 0.63 0.17 0.07 asserted 1.00
s3 To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. method 0.99 0.87 0.04 asserted 1.00
s4 We establish the existence of positive solutions to the scheme. result 0.88 0.96 0.11 asserted 1.00
s5 Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. result 0.83 0.94 0.25 asserted 1.00
s6 Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity. result 0.95 0.96 0.39 asserted 0.99

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.89.

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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

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