An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912712356790272 |
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| author | Guo, Jing Wang, Cheng Wang, Dong |
| author_facet | Guo, Jing Wang, Cheng Wang, Dong |
| contents | A numerical framework is proposed and analyzed for computing the ground state of Bose--Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce the $L^2$ conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, an $L^2$ normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3) $L^2$ norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate Guo, Jing Wang, Cheng Wang, Dong Numerical Analysis 65K10, 65M06, 65M12, 65Z05, 81-08 A numerical framework is proposed and analyzed for computing the ground state of Bose--Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce the $L^2$ conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, an $L^2$ normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3) $L^2$ norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions. |
| title | An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate |
| topic | Numerical Analysis 65K10, 65M06, 65M12, 65Z05, 81-08 |
| url | https://arxiv.org/abs/2511.12411 |