Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws
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| Format: | Preprint |
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2025
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| _version_ | 1866916981849980928 |
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| author | De Nitti, Nicola Huang, Kuang |
| author_facet | De Nitti, Nicola Huang, Kuang |
| contents | We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t ρ_\varepsilon + \partial_x(V(ρ_\varepsilon \ast γ_\varepsilon) ρ_\varepsilon) = 0 $ with a suitable convex kernel $ γ_\varepsilon $, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter $ \varepsilon $ and mesh size $ h $ tend to zero simultaneously, the discrete approximation $ W_{\varepsilon,h} $ of $ W_\varepsilon := ρ_\varepsilon \ast γ_\varepsilon $ converges to the entropy solution of the (local) scalar conservation law $ \partial_t ρ+ \partial_x(V(ρ) ρ) = 0 $, with an explicit convergence rate estimate of order $ \varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t} $. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation $ ρ_{\varepsilon,h} $ of $ ρ_\varepsilon $, along with an $ \mathrm{L}^1 $-contraction property for $ W_\varepsilon $. The key ingredients in proving these results are uniform $ \mathrm{L}^\infty $- and $\mathrm{TV}$-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws De Nitti, Nicola Huang, Kuang Numerical Analysis Analysis of PDEs 35L65 We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t ρ_\varepsilon + \partial_x(V(ρ_\varepsilon \ast γ_\varepsilon) ρ_\varepsilon) = 0 $ with a suitable convex kernel $ γ_\varepsilon $, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter $ \varepsilon $ and mesh size $ h $ tend to zero simultaneously, the discrete approximation $ W_{\varepsilon,h} $ of $ W_\varepsilon := ρ_\varepsilon \ast γ_\varepsilon $ converges to the entropy solution of the (local) scalar conservation law $ \partial_t ρ+ \partial_x(V(ρ) ρ) = 0 $, with an explicit convergence rate estimate of order $ \varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t} $. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation $ ρ_{\varepsilon,h} $ of $ ρ_\varepsilon $, along with an $ \mathrm{L}^1 $-contraction property for $ W_\varepsilon $. The key ingredients in proving these results are uniform $ \mathrm{L}^\infty $- and $\mathrm{TV}$-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution. |
| title | Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws |
| topic | Numerical Analysis Analysis of PDEs 35L65 |
| url | https://arxiv.org/abs/2510.00221 |