A monotone finite volume scheme for linear drift-diffusion and pure drift equations on one-dimensional graphs

Beatrice Crippa; Anna Scotti; Andrea Villa; Beatrice Crippa; Anna Scotti; Andrea Villa

doi:10.3934/nhm.2025029

Networks and Heterogeneous Media

2025, Volume 20, Issue 2: 670-700. doi: 10.3934/nhm.2025029

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Research article

A monotone finite volume scheme for linear drift-diffusion and pure drift equations on one-dimensional graphs

1.
MOX-Laboratory for Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Milan 20133, Italy
2.
Ricerca Sul Sistema Energetico (RSE), Milano 20134, Italy

Received: 28 October 2024 Revised: 28 April 2025 Accepted: 23 May 2025 Published: 12 June 2025

We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending finite volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of electrical treeing.
- drift-diffusion equations,
- transport equations,
- one-dimensional graphs,
- upwind finite volumes,
- electrical treeing
Citation: Beatrice Crippa, Anna Scotti, Andrea Villa. A monotone finite volume scheme for linear drift-diffusion and pure drift equations on one-dimensional graphs[J]. Networks and Heterogeneous Media, 2025, 20(2): 670-700. doi: 10.3934/nhm.2025029

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Abstract

We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending finite volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of electrical treeing.

References

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