Numerical schemes for a class of nonlocal conservation laws: a general approach

Jan Friedrich; Sanjibanee Sudha; Samala Rathan; Jan Friedrich; Sanjibanee Sudha; Samala Rathan

doi:10.3934/nhm.2023058

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1335-1354. doi: 10.3934/nhm.2023058

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Numerical schemes for a class of nonlocal conservation laws: a general approach

1.
RWTH Aachen University, Institute of Applied Mathematics, 52064 Aachen, Germany
2.
Department of Humanities and Sciences, Indian Institute of Petroleum and Energy, Visakhapatnam, Andhra Pradesh, India

Received: 16 February 2023 Revised: 25 March 2023 Accepted: 09 April 2023 Published: 15 May 2023

In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.
- Nonlocal conservation laws,
- monotone schemes,
- traffic flow,
- sedimentation model,
- Finite-volume schemes
Citation: Jan Friedrich, Sanjibanee Sudha, Samala Rathan. Numerical schemes for a class of nonlocal conservation laws: a general approach[J]. Networks and Heterogeneous Media, 2023, 18(3): 1335-1354. doi: 10.3934/nhm.2023058

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Abstract

In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.

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