Conservative modified Crank–Nicolson and time-splitting wavelet methods for modeling Bose–Einstein condensates in delta potentials

摘要

This paper explores two wavelet-based energy-conserving algorithms for the Gross–Pitaevskii equation with delta potentials in Bose–Einstein condensates, named modified Crank–Nicolson wavelet method and time-splitting wavelet method, respectively. Both proposed methods can preserve the intrinsic properties of original problems as much as possible. Meanwhile, the rigorous error estimates and some conservative properties are investigated. They are proved to preserve the charge conservation exactly. The global energy conservation laws can be preserved under several conditions. In practical computations, to avoid a large drift in energy values caused by discontinuous potential well, an improved discrete delta function is implemented. Numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.