Some remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866909979192066048 |
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| author | Zlotnik, Alexander Zlotnik, Ilya |
| author_facet | Zlotnik, Alexander Zlotnik, Ilya |
| contents | We consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step $h$. Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on $h$ and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. This allows us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs (for the first time). Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1406_5102 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Some remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis Zlotnik, Alexander Zlotnik, Ilya Numerical Analysis 65M06, 35Q40 We consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step $h$. Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on $h$ and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. This allows us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs (for the first time). Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed. |
| title | Some remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis |
| topic | Numerical Analysis 65M06, 35Q40 |
| url | https://arxiv.org/abs/1406.5102 |