Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux
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- by Aekta Aggarwal and Ganesh Vaidya;
- Math. Comp.
- DOI: https://doi.org/10.1090/mcom/3976
- Published electronically: May 13, 2024
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Abstract:
We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.References
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Bibliographic Information
- Aekta Aggarwal
- Affiliation: Operations Management And Quantitative Techniques Area, Indian Institute of Management Indore, Rau-Pithampur Road, Indore, Madhya Pradesh 453556, India
- MR Author ID: 1103032
- ORCID: 0000-0002-0710-6647
- Email: aektaaggarwal@iimidr.ac.in
- Ganesh Vaidya
- Affiliation: Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
- MR Author ID: 1402927
- Email: gmv5228@psu.edu
- Received by editor(s): January 24, 2023
- Received by editor(s) in revised form: July 21, 2023, November 13, 2023, March 26, 2024, and April 2, 2024
- Published electronically: May 13, 2024
- Additional Notes: Parts of this work were carried out during the second author’s tenure of the ERCIM ‘Alain Bensoussan’ Fellowship Programme, and were supported in part by the project IMod — Partial differential equations, statistics and data: An interdisciplinary approach to data-based modelling, project number 325114, from the Research Council of Norway. Parts of this work were also carried out during the second author’s tenure at IIM Indore and the first author’s research visit at NTNU, and were partially supported by the first author’s Seed Money Grant SM/11/2021/22 and by the first author’s Faculty Development Allowance, from IIM Indore.
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp.
- MSC (2020): Primary 35L65, 35B44, 35A01, 65M06, 65M08
- DOI: https://doi.org/10.1090/mcom/3976