| Issue |
ESAIM: M2AN
Volume 57, Number 6, November-December 2023
|
|
|---|---|---|
| Page(s) | 3439 - 3481 | |
| DOI | https://doi.org/10.1051/m2an/2023080 | |
| Published online | 29 November 2023 | |
Convergence of a second-order scheme for non-local conservation laws
1
Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore 560065, India
2
School of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, India
* Corresponding author: sudarshan@iisertvm.ac.in
Received:
17
May
2023
Accepted:
28
September
2023
In this article, we present the convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws. We combine a MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme. The resulting scheme is shown to converge to a weak solution by establishing the maximum principle, bounded variation estimates and L1 Lipschitz continuity in time. Further, using a space-step dependent slope limiter, we prove its convergence to the entropy solution. We also propose a MUSCL-Hancock type second-order scheme which requires only one intermediate stage unlike the Runge-Kutta schemes and is easier to implement. The performance of the proposed second-order schemes in comparison to a first-order scheme is demonstrated through several numerical experiments.
Mathematics Subject Classification: 35L65 / 76A30 / 65M08 / 65M12
Key words: Non-local conservation laws / MUSCL method / second-order scheme / MUSCL-Hancock scheme / convergence analysis / entropy solution
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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