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- Title
Compact exponential energy-preserving method for the Schrödinger equation in the semiclassical regime.
- Authors
Chen, Juan; Cai, Jixiang
- Abstract
We develop an efficient energy-preserving exponential numerical scheme for the Schrödinger equation in the semiclassical limit with small parameter $ \varepsilon $. The scheme is based on adopting the Fourier pseudo-spectral method in space and applying the compact implicit integrating factor method with the discrete gradient method in time. The compact representation of our scheme saves the storage requirement and operation count. Ample numerical tests for the equation in one dimension suggest that, in order to compute correct physical quantities, the meshing strategies are: $ \tau = \mathcal{O}(\varepsilon) $ and $ h = \mathcal{O}(\varepsilon) $ for the nonlinear equation with strong $ \mathcal{O}(1) $–defocusing or weak $ \mathcal{O}(\varepsilon) $–focusing nonlinearities, and $ \tau $ independent of $ \varepsilon $ and $ h = \mathcal{O}(\varepsilon) $ for the linear equation or nonlinear equation with weak $ \mathcal{O}(\varepsilon) $–focusing nonlinearities. Numerical experiments for the two-/three-dimensional equation show the power of our scheme in simulating Bose-Einstein condensation.
- Subjects
NONLINEAR equations; SEPARATION of variables; SEMICLASSICAL limits; COMPACT spaces (Topology); LINEAR equations
- Publication
Discrete & Continuous Dynamical Systems - Series B, 2024, Vol 29, Issue 5, p1
- ISSN
1531-3492
- Publication type
Article
- DOI
10.3934/dcdsb.2023170