Low regularity error estimates for the time integration of 2D NLS
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917086860673024 |
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| author | Ji, Lun Ostermann, Alexander Rousset, Frédéric Schratz, Katharina |
| author_facet | Ji, Lun Ostermann, Alexander Rousset, Frédéric Schratz, Katharina |
| contents | A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $τ^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10639 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Low regularity error estimates for the time integration of 2D NLS Ji, Lun Ostermann, Alexander Rousset, Frédéric Schratz, Katharina Numerical Analysis 65M12, 65M15, 35Q55 A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $τ^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp. |
| title | Low regularity error estimates for the time integration of 2D NLS |
| topic | Numerical Analysis 65M12, 65M15, 35Q55 |
| url | https://arxiv.org/abs/2301.10639 |