Conservation laws with discontinuous flux function on networks: a splitting algorithm

Jan Friedrich; Simone Göttlich; Annika Uphoff; Jan Friedrich; Simone Göttlich; Annika Uphoff

doi:10.3934/nhm.2023001

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 1-28. doi: 10.3934/nhm.2023001

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Research article

Conservation laws with discontinuous flux function on networks: a splitting algorithm

1.
RWTH Aachen University, Institute of Applied Mathematics, 52064 Aachen, Germany
2.
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

Received: 08 April 2022 Revised: 24 August 2022 Accepted: 03 September 2022 Published: 14 October 2022

In this article, we present an extension of the splitting algorithm proposed in [22] to networks of conservation laws with piecewise linear discontinuous flux functions in the unknown. We start with the discussion of a suitable Riemann solver at the junction and then describe a strategy how to use the splitting algorithm on the network. In particular, we focus on two types of junctions, i.e., junctions where the number of outgoing roads does not exceed the number of incoming roads (dispersing type) and junctions with two incoming and one outgoing road (merging type). Finally, numerical examples demonstrate the accuracy of the splitting algorithm by comparisons to the exact solution and other approaches used in the literature.
- traffic flow networks,
- LWR model,
- conservation laws with discontinuous flux,
- numerical schemes,
- splitting algorithm
Citation: Jan Friedrich, Simone Göttlich, Annika Uphoff. Conservation laws with discontinuous flux function on networks: a splitting algorithm[J]. Networks and Heterogeneous Media, 2023, 18(1): 1-28. doi: 10.3934/nhm.2023001

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Abstract

In this article, we present an extension of the splitting algorithm proposed in [22] to networks of conservation laws with piecewise linear discontinuous flux functions in the unknown. We start with the discussion of a suitable Riemann solver at the junction and then describe a strategy how to use the splitting algorithm on the network. In particular, we focus on two types of junctions, i.e., junctions where the number of outgoing roads does not exceed the number of incoming roads (dispersing type) and junctions with two incoming and one outgoing road (merging type). Finally, numerical examples demonstrate the accuracy of the splitting algorithm by comparisons to the exact solution and other approaches used in the literature.

References

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