A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2013
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| _version_ | 1866908741895454720 |
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| author | Ducomet, Bernard Zlotnik, Alexander Romanova, Alla |
| author_facet | Ducomet, Bernard Zlotnik, Alexander Romanova, Alla |
| contents | An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1309_7280 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped Ducomet, Bernard Zlotnik, Alexander Romanova, Alla Numerical Analysis Computational Physics Quantum Physics 65M06, 65M12, 35Q40 An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included. |
| title | A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped |
| topic | Numerical Analysis Computational Physics Quantum Physics 65M06, 65M12, 35Q40 |
| url | https://arxiv.org/abs/1309.7280 |