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On the transport limit of singularly perturbed convection–diffusion problems on networks

Egger, Herbert ; Philippi, Nora (2024)
On the transport limit of singularly perturbed convection–diffusion problems on networks.
In: Mathematical Methods in the Applied Sciences, 2021, 44 (6)
doi: 10.26083/tuprints-00017804
Artikel, Zweitveröffentlichung, Verlagsversion

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Kurzbeschreibung (Abstract)

We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order O(ϵ) in the L∞(L²)‐norm with diffusion ϵ → 0. In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

Typ des Eintrags: Artikel
Erschienen: 2024
Autor(en): Egger, Herbert ; Philippi, Nora
Art des Eintrags: Zweitveröffentlichung
Titel: On the transport limit of singularly perturbed convection–diffusion problems on networks
Sprache: Englisch
Publikationsjahr: 12 Februar 2024
Ort: Darmstadt
Publikationsdatum der Erstveröffentlichung: 2021
Ort der Erstveröffentlichung: Chichester
Verlag: John Wiley & Sons
Titel der Zeitschrift, Zeitung oder Schriftenreihe: Mathematical Methods in the Applied Sciences
Jahrgang/Volume einer Zeitschrift: 44
(Heft-)Nummer: 6
DOI: 10.26083/tuprints-00017804
URL / URN: https://tuprints.ulb.tu-darmstadt.de/17804
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Herkunft: Zweitveröffentlichung DeepGreen
Kurzbeschreibung (Abstract):

We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order O(ϵ) in the L∞(L²)‐norm with diffusion ϵ → 0. In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

Freie Schlagworte: asymptotic analysis, diffusion and convection (76R05), partial differential equations on networks, singular perturbations in the context of PDEs (35B25)
Status: Verlagsversion
URN: urn:nbn:de:tuda-tuprints-178044
Zusätzliche Informationen:

MSC CLASSIFICATION: 35B25; 35K20; 35R02; 76M45

Sachgruppe der Dewey Dezimalklassifikatin (DDC): 500 Naturwissenschaften und Mathematik > 510 Mathematik
Fachbereich(e)/-gebiet(e): 04 Fachbereich Mathematik
04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen
Hinterlegungsdatum: 12 Feb 2024 13:46
Letzte Änderung: 13 Feb 2024 14:55
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