A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915060417298432 |
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| author | Abreu, Eduardo Chiri, Maria Teresa De la cruz, Richard Juajibioy, Juan Lambert, Wanderson |
| author_facet | Abreu, Eduardo Chiri, Maria Teresa De la cruz, Richard Juajibioy, Juan Lambert, Wanderson |
| contents | In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation $u_t + (k(x)u(1-u))_x = 0$. This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math. 406 (2022) 114011] for scalar conservation laws. We derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at $x=0$, we can provide a convergence proof for our scheme in $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ by using Helly's compactness theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux Abreu, Eduardo Chiri, Maria Teresa De la cruz, Richard Juajibioy, Juan Lambert, Wanderson Numerical Analysis 35L50, 35L65, 35R05, 76S05 In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation $u_t + (k(x)u(1-u))_x = 0$. This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math. 406 (2022) 114011] for scalar conservation laws. We derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at $x=0$, we can provide a convergence proof for our scheme in $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ by using Helly's compactness theorem. |
| title | A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux |
| topic | Numerical Analysis 35L50, 35L65, 35R05, 76S05 |
| url | https://arxiv.org/abs/2412.06692 |