A numerical view on α-dissipative solutions of the Hunter-Saxton equation
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915108578394112 |
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| author | Christiansen, Thomas Grunert, Katrin |
| author_facet | Christiansen, Thomas Grunert, Katrin |
| contents | We propose a new numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α$ belongs to $W^{1, \infty}(\mathbb{R}, [0, 1))$. The method combines a projection operator with a generalized method of characteristics and an iteration scheme, which is based on enforcing minimal time steps whenever breaking times cluster. Numerical examples illustrate that these minimal time steps increase the efficiency of the algorithm substantially. Moreover, convergence of the wave profile is shown in $C([0, T], L^{\infty}(\mathbb{R}))$ for any finite $T \geq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A numerical view on α-dissipative solutions of the Hunter-Saxton equation Christiansen, Thomas Grunert, Katrin Numerical Analysis Analysis of PDEs 65M12, 65M25 (Primary) 65M06, 35Q35 (Secondary) We propose a new numerical method for $α$-dissipative solutions of the Hunter-Saxton equation, where $α$ belongs to $W^{1, \infty}(\mathbb{R}, [0, 1))$. The method combines a projection operator with a generalized method of characteristics and an iteration scheme, which is based on enforcing minimal time steps whenever breaking times cluster. Numerical examples illustrate that these minimal time steps increase the efficiency of the algorithm substantially. Moreover, convergence of the wave profile is shown in $C([0, T], L^{\infty}(\mathbb{R}))$ for any finite $T \geq 0$. |
| title | A numerical view on α-dissipative solutions of the Hunter-Saxton equation |
| topic | Numerical Analysis Analysis of PDEs 65M12, 65M25 (Primary) 65M06, 35Q35 (Secondary) |
| url | https://arxiv.org/abs/2404.11174 |