The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2013
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| _version_ | 1866917178649870336 |
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| author | Ducomet, Bernard Zlotnik, Alexander Zlotnik, Ilya |
| author_facet | Ducomet, Bernard Zlotnik, Alexander Zlotnik, Ilya |
| contents | We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1303_3471 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip Ducomet, Bernard Zlotnik, Alexander Zlotnik, Ilya Numerical Analysis Computational Physics Quantum Physics 65M06, 65M12, 35Q40 We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method. |
| title | The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip |
| topic | Numerical Analysis Computational Physics Quantum Physics 65M06, 65M12, 35Q40 |
| url | https://arxiv.org/abs/1303.3471 |