Comparison of two different integration methods for the (1+1)-Dimensional Schrödinger-Poisson Equation
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929338172047360 |
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| author | Schwersenz, Nico Loaiza, Victor Zimmermann, Tim Madroñero, Javier Wimberger, Sandro |
| author_facet | Schwersenz, Nico Loaiza, Victor Zimmermann, Tim Madroñero, Javier Wimberger, Sandro |
| contents | We compare two different numerical methods to integrate in time spatially delocalized initial densities using the Schrödinger-Poisson equation system as the evolution law. The basic equation is a nonlinear Schrödinger equation with an auto-gravitating potential created by the wave function density itself. The latter is determined as a solution of Poisson's equation modelling, e.g., non-relativistic gravity. For reasons of complexity, we treat a one-dimensional version of the problem whose numerical integration is still challenging because of the extreme long-range forces (being constant in the asymptotic limit). Both of our methods, a Strang splitting scheme and a basis function approach using B-splines, are compared in numerical convergence and effectivity. Overall, our Strang-splitting evolution compares favourably with the B-spline method. In particular, by using an adaptive time-stepper rather large one-dimensional boxes can be treated. These results give hope for extensions to two spatial dimensions for not too small boxes and large evolution times necessary for describing, for instance, dark matter formation over cosmologically relevant scales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison of two different integration methods for the (1+1)-Dimensional Schrödinger-Poisson Equation Schwersenz, Nico Loaiza, Victor Zimmermann, Tim Madroñero, Javier Wimberger, Sandro General Relativity and Quantum Cosmology Quantum Gases Computational Physics We compare two different numerical methods to integrate in time spatially delocalized initial densities using the Schrödinger-Poisson equation system as the evolution law. The basic equation is a nonlinear Schrödinger equation with an auto-gravitating potential created by the wave function density itself. The latter is determined as a solution of Poisson's equation modelling, e.g., non-relativistic gravity. For reasons of complexity, we treat a one-dimensional version of the problem whose numerical integration is still challenging because of the extreme long-range forces (being constant in the asymptotic limit). Both of our methods, a Strang splitting scheme and a basis function approach using B-splines, are compared in numerical convergence and effectivity. Overall, our Strang-splitting evolution compares favourably with the B-spline method. In particular, by using an adaptive time-stepper rather large one-dimensional boxes can be treated. These results give hope for extensions to two spatial dimensions for not too small boxes and large evolution times necessary for describing, for instance, dark matter formation over cosmologically relevant scales. |
| title | Comparison of two different integration methods for the (1+1)-Dimensional Schrödinger-Poisson Equation |
| topic | General Relativity and Quantum Cosmology Quantum Gases Computational Physics |
| url | https://arxiv.org/abs/2405.04924 |