Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas
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arXiv
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| Format: | Preprint |
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2024
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| author | Bronsard, Yvonne Alama Chen, Xi Dolbeault, Matthieu |
| author_facet | Bronsard, Yvonne Alama Chen, Xi Dolbeault, Matthieu |
| contents | We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szegő equations on the torus, which are $\textit{exact in time}$ with $\textit{spectral accuracy}$ in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ in $L^2$ norm for initial data in $H^s(\mathbb T)$, $s>1$, with an error constant depending $\textit{linearly}$ on the final time instead of exponentially. These schemes are based on $\textit{explicit formulas}$, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas Bronsard, Yvonne Alama Chen, Xi Dolbeault, Matthieu Numerical Analysis Analysis of PDEs Spectral Theory 37K10, 65M70, 65M15, 37K15, 35Q35 We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szegő equations on the torus, which are $\textit{exact in time}$ with $\textit{spectral accuracy}$ in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ in $L^2$ norm for initial data in $H^s(\mathbb T)$, $s>1$, with an error constant depending $\textit{linearly}$ on the final time instead of exponentially. These schemes are based on $\textit{explicit formulas}$, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations. |
| title | Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas |
| topic | Numerical Analysis Analysis of PDEs Spectral Theory 37K10, 65M70, 65M15, 37K15, 35Q35 |
| url | https://arxiv.org/abs/2412.13480 |