Central schemes for networked scalar conservation laws

Michael Herty; Niklas Kolbe; Siegfried Müller; Michael Herty; Niklas Kolbe; Siegfried Müller

doi:10.3934/nhm.2023012

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 310-340. doi: 10.3934/nhm.2023012

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Research article

Central schemes for networked scalar conservation laws

Institute of Geometry and Practical Mathematics, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

Received: 12 September 2022 Revised: 07 November 2022 Accepted: 30 November 2022 Published: 22 December 2022

We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure. Appl. Math. 48 (1995), 235-276] and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative and yields well defined and easy-to-compute coupling conditions even for general networks. We discuss higher order extension of the scheme and applications to traffic flow and two-phase flow. In the former we compare with results obtained in literature.
- Coupled conservation laws,
- finite-volume schemes,
- coupling conditions,
- relaxation system,
- scalar conservation laws
Citation: Michael Herty, Niklas Kolbe, Siegfried Müller. Central schemes for networked scalar conservation laws[J]. Networks and Heterogeneous Media, 2023, 18(1): 310-340. doi: 10.3934/nhm.2023012

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Abstract

We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure. Appl. Math. 48 (1995), 235-276] and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative and yields well defined and easy-to-compute coupling conditions even for general networks. We discuss higher order extension of the scheme and applications to traffic flow and two-phase flow. In the former we compare with results obtained in literature.

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