A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918012649472000 |
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| author | Christiansen, Thomas Grunert, Katrin Nordli, Anders Solem, Susanne |
| author_facet | Christiansen, Thomas Grunert, Katrin Nordli, Anders Solem, Susanne |
| contents | A convergent numerical method for $α$-dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in $L^{\infty}$ for all $t \geq 0$, while a subsequence of the energy density converges weakly for almost every time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_08763 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation Christiansen, Thomas Grunert, Katrin Nordli, Anders Solem, Susanne Numerical Analysis Analysis of PDEs 65M12, 65M25 (Primary) 65M06, 35Q35 (Secondary) A convergent numerical method for $α$-dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in $L^{\infty}$ for all $t \geq 0$, while a subsequence of the energy density converges weakly for almost every time. |
| title | A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation |
| topic | Numerical Analysis Analysis of PDEs 65M12, 65M25 (Primary) 65M06, 35Q35 (Secondary) |
| url | https://arxiv.org/abs/2303.08763 |