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A normalized gradient flow method with attractive-repulsive splitting for computing ground states of Bose-Einstein condensates with higher-order interaction. (English) Zbl 1415.65198
Summary: In this paper, we generalize the normalized gradient flow method to compute the ground states of Bose-Einstein condensates (BEC) with higher order interactions (HOI), which is modeled via the modified Gross-Pitaevskii equation (MGPE). Schemes constructed in naive ways suffer from severe stability problems due to the high restrictions on time steps. To build an efficient and stable scheme, we split the HOI term into two parts with each part treated separately. The part corresponding to a repulsive/positive energy is treated semi-implicitly while the one corresponding to an attractive/negative energy is treated fully explicitly. Based on the splitting, we construct the BEFD-splitting and BESP-splitting schemes. A variety of numerical experiments show that the splitting will improve the stability of the schemes significantly. Besides, we will show that the methods can be applied to multidimensional problems and to the computation of the first excited state as well.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
81V70 Many-body theory; quantum Hall effect
35Q40 PDEs in connection with quantum mechanics
Software:
GPELab; Wesseling
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