Egger, Herbert ; Philippi, Nora (2021)
On the transport limit of singularly perturbed convection–diffusion problems on networks.
In: Mathematical Methods in the Applied Sciences, 44 (6)
doi: 10.1002/mma.7084
Artikel, Bibliographie
Dies ist die neueste Version dieses Eintrags.
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: | 2021 |
| Autor(en): | Egger, Herbert ; Philippi, Nora |
| Art des Eintrags: | Bibliographie |
| Titel: | On the transport limit of singularly perturbed convection–diffusion problems on networks |
| Sprache: | Englisch |
| Publikationsjahr: | 2021 |
| Ort: | 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: | 10.1002/mma.7084 |
| Zugehörige Links: | |
| 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) |
| 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: | 13 Feb 2024 14:56 |
| Letzte Änderung: | 13 Feb 2024 14:56 |
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Verfügbare Versionen dieses Eintrags
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On the transport limit of singularly perturbed convection–diffusion problems on networks. (deposited 12 Feb 2024 13:46)
- On the transport limit of singularly perturbed convection–diffusion problems on networks. (deposited 13 Feb 2024 14:56) [Gegenwärtig angezeigt]
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