A finite difference continuation method for computing energy levels of Bose–Einstein condensates

We study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose–Einstein condensates (BEC), which is governed by the Gross–Pitaevskii equation (GPE). We choose the chemical potential λ   as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrödinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as well as excited-states of the SEP in the variational form. We also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(ht)O(ht) in the discrete H1H1 norm is achieved, where t=2t=2 and t=1.5t=1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential.