On the convergence rate of a numerical method for the Hunter-Saxton equation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914966957719552 |
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| author | Christiansen, Thomas |
| author_facet | Christiansen, Thomas |
| contents | We derive a robust error estimate for a recently proposed numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α\in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ 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 $h \in (0, 2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(Δx^{\fracβ{8}})$ in $L^{\infty}(\mathbb{R})$.
Moreover, if $α=0$, then the rate improves to $O(Δx^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(Δx^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convergence rate of a numerical method for the Hunter-Saxton equation Christiansen, Thomas Numerical Analysis Analysis of PDEs 65M15, 65M12, 65M25 We derive a robust error estimate for a recently proposed numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α\in [0, 1]$. In particular, if the following two conditions hold: i) there exist a constant $C > 0$ 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 $h \in (0, 2]$, and ii), the singular continuous part of the initial energy measure is zero, then the numerical wave profile converges with order $O(Δx^{\fracβ{8}})$ in $L^{\infty}(\mathbb{R})$. Moreover, if $α=0$, then the rate improves to $O(Δx^{\frac{1}{4}})$ without the above assumptions, and we also obtain a convergence rate for the associated energy measure - it converges with order $O(Δx^{\frac{1}{2}})$ in the bounded Lipschitz metric. These convergence rates are illustrated by several examples. |
| title | On the convergence rate of a numerical method for the Hunter-Saxton equation |
| topic | Numerical Analysis Analysis of PDEs 65M15, 65M12, 65M25 |
| url | https://arxiv.org/abs/2409.18903 |