Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914092722159616 |
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| author | Christiansen, Thomas Grunert, Katrin |
| author_facet | Christiansen, Thomas Grunert, Katrin |
| contents | We prove that $α$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $α\in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(Δx^{{1}/{8}}+Δx^{β/{4}})$ in $L^{\infty}(\mathbb{R})$, provided there exist constants $C > 0$ and $β\in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^β$ for all $h \in (0, 2]$. The derived convergence rate is exemplified by a number of numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07712 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation Christiansen, Thomas Grunert, Katrin Numerical Analysis Analysis of PDEs 65M15, 65M25 (Primary) 35Q35 (Secondary) We prove that $α$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $α\in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(Δx^{{1}/{8}}+Δx^{β/{4}})$ in $L^{\infty}(\mathbb{R})$, provided there exist constants $C > 0$ and $β\in (0, 1]$ such that the initial spatial derivative $\bar{u}_{x}$ satisfies $\|\bar{u}_x(\cdot + h) - \bar{u}_x(\cdot)\|_2 \leq Ch^β$ for all $h \in (0, 2]$. The derived convergence rate is exemplified by a number of numerical experiments. |
| title | Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation |
| topic | Numerical Analysis Analysis of PDEs 65M15, 65M25 (Primary) 35Q35 (Secondary) |
| url | https://arxiv.org/abs/2411.07712 |